{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Simulations and Statistical Inference"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Registered S3 methods overwritten by 'ggplot2':\n",
      "  method         from \n",
      "  [.quosures     rlang\n",
      "  c.quosures     rlang\n",
      "  print.quosures rlang\n",
      "── \u001b[1mAttaching packages\u001b[22m ─────────────────────────────────────── tidyverse 1.2.1 ──\n",
      "\u001b[32m✔\u001b[39m \u001b[34mggplot2\u001b[39m 3.1.1     \u001b[32m✔\u001b[39m \u001b[34mpurrr  \u001b[39m 0.3.2\n",
      "\u001b[32m✔\u001b[39m \u001b[34mtibble \u001b[39m 2.1.2     \u001b[32m✔\u001b[39m \u001b[34mdplyr  \u001b[39m 0.8.1\n",
      "\u001b[32m✔\u001b[39m \u001b[34mtidyr  \u001b[39m 0.8.3     \u001b[32m✔\u001b[39m \u001b[34mstringr\u001b[39m 1.4.0\n",
      "\u001b[32m✔\u001b[39m \u001b[34mreadr  \u001b[39m 1.3.1     \u001b[32m✔\u001b[39m \u001b[34mforcats\u001b[39m 0.4.0\n",
      "── \u001b[1mConflicts\u001b[22m ────────────────────────────────────────── tidyverse_conflicts() ──\n",
      "\u001b[31m✖\u001b[39m \u001b[34mdplyr\u001b[39m::\u001b[32mfilter()\u001b[39m masks \u001b[34mstats\u001b[39m::filter()\n",
      "\u001b[31m✖\u001b[39m \u001b[34mdplyr\u001b[39m::\u001b[32mlag()\u001b[39m    masks \u001b[34mstats\u001b[39m::lag()\n"
     ]
    }
   ],
   "source": [
    "library(tidyverse)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "options(repr.plot.width=4, repr.plot.height=3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Set random number seed for reproducibility"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "set.seed(42)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Generating numbers"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using the `c` function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>1</li>\n",
       "\t<li>2</li>\n",
       "\t<li>3</li>\n",
       "\t<li>5</li>\n",
       "\t<li>8</li>\n",
       "\t<li>13</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 1\n",
       "\\item 2\n",
       "\\item 3\n",
       "\\item 5\n",
       "\\item 8\n",
       "\\item 13\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 1\n",
       "2. 2\n",
       "3. 3\n",
       "4. 5\n",
       "5. 8\n",
       "6. 13\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1]  1  2  3  5  8 13"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "c(1,2,3,5,8,13)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using the `:` operator"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>1</li>\n",
       "\t<li>2</li>\n",
       "\t<li>3</li>\n",
       "\t<li>4</li>\n",
       "\t<li>5</li>\n",
       "\t<li>6</li>\n",
       "\t<li>7</li>\n",
       "\t<li>8</li>\n",
       "\t<li>9</li>\n",
       "\t<li>10</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 1\n",
       "\\item 2\n",
       "\\item 3\n",
       "\\item 4\n",
       "\\item 5\n",
       "\\item 6\n",
       "\\item 7\n",
       "\\item 8\n",
       "\\item 9\n",
       "\\item 10\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 1\n",
       "2. 2\n",
       "3. 3\n",
       "4. 4\n",
       "5. 5\n",
       "6. 6\n",
       "7. 7\n",
       "8. 8\n",
       "9. 9\n",
       "10. 10\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       " [1]  1  2  3  4  5  6  7  8  9 10"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "1:10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>10</li>\n",
       "\t<li>9</li>\n",
       "\t<li>8</li>\n",
       "\t<li>7</li>\n",
       "\t<li>6</li>\n",
       "\t<li>5</li>\n",
       "\t<li>4</li>\n",
       "\t<li>3</li>\n",
       "\t<li>2</li>\n",
       "\t<li>1</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 10\n",
       "\\item 9\n",
       "\\item 8\n",
       "\\item 7\n",
       "\\item 6\n",
       "\\item 5\n",
       "\\item 4\n",
       "\\item 3\n",
       "\\item 2\n",
       "\\item 1\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 10\n",
       "2. 9\n",
       "3. 8\n",
       "4. 7\n",
       "5. 6\n",
       "6. 5\n",
       "7. 4\n",
       "8. 3\n",
       "9. 2\n",
       "10. 1\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       " [1] 10  9  8  7  6  5  4  3  2  1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "10:1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using the `seq` function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>1</li>\n",
       "\t<li>2</li>\n",
       "\t<li>3</li>\n",
       "\t<li>4</li>\n",
       "\t<li>5</li>\n",
       "\t<li>6</li>\n",
       "\t<li>7</li>\n",
       "\t<li>8</li>\n",
       "\t<li>9</li>\n",
       "\t<li>10</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 1\n",
       "\\item 2\n",
       "\\item 3\n",
       "\\item 4\n",
       "\\item 5\n",
       "\\item 6\n",
       "\\item 7\n",
       "\\item 8\n",
       "\\item 9\n",
       "\\item 10\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 1\n",
       "2. 2\n",
       "3. 3\n",
       "4. 4\n",
       "5. 5\n",
       "6. 6\n",
       "7. 7\n",
       "8. 8\n",
       "9. 9\n",
       "10. 10\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       " [1]  1  2  3  4  5  6  7  8  9 10"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "seq(1, 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>1</li>\n",
       "\t<li>3</li>\n",
       "\t<li>5</li>\n",
       "\t<li>7</li>\n",
       "\t<li>9</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 1\n",
       "\\item 3\n",
       "\\item 5\n",
       "\\item 7\n",
       "\\item 9\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 1\n",
       "2. 3\n",
       "3. 5\n",
       "4. 7\n",
       "5. 9\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 1 3 5 7 9"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "seq(1, 10, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>10</li>\n",
       "\t<li>9</li>\n",
       "\t<li>8</li>\n",
       "\t<li>7</li>\n",
       "\t<li>6</li>\n",
       "\t<li>5</li>\n",
       "\t<li>4</li>\n",
       "\t<li>3</li>\n",
       "\t<li>2</li>\n",
       "\t<li>1</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 10\n",
       "\\item 9\n",
       "\\item 8\n",
       "\\item 7\n",
       "\\item 6\n",
       "\\item 5\n",
       "\\item 4\n",
       "\\item 3\n",
       "\\item 2\n",
       "\\item 1\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 10\n",
       "2. 9\n",
       "3. 8\n",
       "4. 7\n",
       "5. 6\n",
       "6. 5\n",
       "7. 4\n",
       "8. 3\n",
       "9. 2\n",
       "10. 1\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       " [1] 10  9  8  7  6  5  4  3  2  1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "seq(10, 1, -1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>0</li>\n",
       "\t<li>0.1</li>\n",
       "\t<li>0.2</li>\n",
       "\t<li>0.3</li>\n",
       "\t<li>0.4</li>\n",
       "\t<li>0.5</li>\n",
       "\t<li>0.6</li>\n",
       "\t<li>0.7</li>\n",
       "\t<li>0.8</li>\n",
       "\t<li>0.9</li>\n",
       "\t<li>1</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 0\n",
       "\\item 0.1\n",
       "\\item 0.2\n",
       "\\item 0.3\n",
       "\\item 0.4\n",
       "\\item 0.5\n",
       "\\item 0.6\n",
       "\\item 0.7\n",
       "\\item 0.8\n",
       "\\item 0.9\n",
       "\\item 1\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 0\n",
       "2. 0.1\n",
       "3. 0.2\n",
       "4. 0.3\n",
       "5. 0.4\n",
       "6. 0.5\n",
       "7. 0.6\n",
       "8. 0.7\n",
       "9. 0.8\n",
       "10. 0.9\n",
       "11. 1\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       " [1] 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "seq(0, 1, length.out = 11)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Using `rep`\n",
    "\n",
    "`rep` is useful for generating repeating data patterns."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>3</li>\n",
       "\t<li>3</li>\n",
       "\t<li>3</li>\n",
       "\t<li>3</li>\n",
       "\t<li>3</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 3\n",
       "\\item 3\n",
       "\\item 3\n",
       "\\item 3\n",
       "\\item 3\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 3\n",
       "2. 3\n",
       "3. 3\n",
       "4. 3\n",
       "5. 3\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 3 3 3 3 3"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rep(3, 5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>1</li>\n",
       "\t<li>2</li>\n",
       "\t<li>3</li>\n",
       "\t<li>1</li>\n",
       "\t<li>2</li>\n",
       "\t<li>3</li>\n",
       "\t<li>1</li>\n",
       "\t<li>2</li>\n",
       "\t<li>3</li>\n",
       "\t<li>1</li>\n",
       "\t<li>2</li>\n",
       "\t<li>3</li>\n",
       "\t<li>1</li>\n",
       "\t<li>2</li>\n",
       "\t<li>3</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 1\n",
       "\\item 2\n",
       "\\item 3\n",
       "\\item 1\n",
       "\\item 2\n",
       "\\item 3\n",
       "\\item 1\n",
       "\\item 2\n",
       "\\item 3\n",
       "\\item 1\n",
       "\\item 2\n",
       "\\item 3\n",
       "\\item 1\n",
       "\\item 2\n",
       "\\item 3\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 1\n",
       "2. 2\n",
       "3. 3\n",
       "4. 1\n",
       "5. 2\n",
       "6. 3\n",
       "7. 1\n",
       "8. 2\n",
       "9. 3\n",
       "10. 1\n",
       "11. 2\n",
       "12. 3\n",
       "13. 1\n",
       "14. 2\n",
       "15. 3\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       " [1] 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rep(1:3, times=5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>1</li>\n",
       "\t<li>1</li>\n",
       "\t<li>1</li>\n",
       "\t<li>1</li>\n",
       "\t<li>1</li>\n",
       "\t<li>2</li>\n",
       "\t<li>2</li>\n",
       "\t<li>2</li>\n",
       "\t<li>2</li>\n",
       "\t<li>2</li>\n",
       "\t<li>3</li>\n",
       "\t<li>3</li>\n",
       "\t<li>3</li>\n",
       "\t<li>3</li>\n",
       "\t<li>3</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 1\n",
       "\\item 1\n",
       "\\item 1\n",
       "\\item 1\n",
       "\\item 1\n",
       "\\item 2\n",
       "\\item 2\n",
       "\\item 2\n",
       "\\item 2\n",
       "\\item 2\n",
       "\\item 3\n",
       "\\item 3\n",
       "\\item 3\n",
       "\\item 3\n",
       "\\item 3\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 1\n",
       "2. 1\n",
       "3. 1\n",
       "4. 1\n",
       "5. 1\n",
       "6. 2\n",
       "7. 2\n",
       "8. 2\n",
       "9. 2\n",
       "10. 2\n",
       "11. 3\n",
       "12. 3\n",
       "13. 3\n",
       "14. 3\n",
       "15. 3\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       " [1] 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rep(1:3, each=5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>1</li>\n",
       "\t<li>2</li>\n",
       "\t<li>3</li>\n",
       "\t<li>1</li>\n",
       "\t<li>2</li>\n",
       "\t<li>3</li>\n",
       "\t<li>1</li>\n",
       "\t<li>2</li>\n",
       "\t<li>3</li>\n",
       "\t<li>1</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 1\n",
       "\\item 2\n",
       "\\item 3\n",
       "\\item 1\n",
       "\\item 2\n",
       "\\item 3\n",
       "\\item 1\n",
       "\\item 2\n",
       "\\item 3\n",
       "\\item 1\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 1\n",
       "2. 2\n",
       "3. 3\n",
       "4. 1\n",
       "5. 2\n",
       "6. 3\n",
       "7. 1\n",
       "8. 2\n",
       "9. 3\n",
       "10. 1\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       " [1] 1 2 3 1 2 3 1 2 3 1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rep(1:3, length.out=10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Using `sample`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Simulate `n` tooses of a coin"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "n <- 10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>'H'</li>\n",
       "\t<li>'H'</li>\n",
       "\t<li>'H'</li>\n",
       "\t<li>'H'</li>\n",
       "\t<li>'T'</li>\n",
       "\t<li>'T'</li>\n",
       "\t<li>'T'</li>\n",
       "\t<li>'T'</li>\n",
       "\t<li>'H'</li>\n",
       "\t<li>'T'</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 'H'\n",
       "\\item 'H'\n",
       "\\item 'H'\n",
       "\\item 'H'\n",
       "\\item 'T'\n",
       "\\item 'T'\n",
       "\\item 'T'\n",
       "\\item 'T'\n",
       "\\item 'H'\n",
       "\\item 'T'\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 'H'\n",
       "2. 'H'\n",
       "3. 'H'\n",
       "4. 'H'\n",
       "5. 'T'\n",
       "6. 'T'\n",
       "7. 'T'\n",
       "8. 'T'\n",
       "9. 'H'\n",
       "10. 'T'\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       " [1] \"H\" \"H\" \"H\" \"H\" \"T\" \"T\" \"T\" \"T\" \"H\" \"T\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t1 <- sample(c('H', 'T'), n, replace = TRUE)\n",
    "t1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "t1\n",
       "H T \n",
       "5 5 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "table(t1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Simulate `n` tosses of a biased coin"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>'T'</li>\n",
       "\t<li>'H'</li>\n",
       "\t<li>'H'</li>\n",
       "\t<li>'T'</li>\n",
       "\t<li>'T'</li>\n",
       "\t<li>'H'</li>\n",
       "\t<li>'H'</li>\n",
       "\t<li>'T'</li>\n",
       "\t<li>'T'</li>\n",
       "\t<li>'T'</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 'T'\n",
       "\\item 'H'\n",
       "\\item 'H'\n",
       "\\item 'T'\n",
       "\\item 'T'\n",
       "\\item 'H'\n",
       "\\item 'H'\n",
       "\\item 'T'\n",
       "\\item 'T'\n",
       "\\item 'T'\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 'T'\n",
       "2. 'H'\n",
       "3. 'H'\n",
       "4. 'T'\n",
       "5. 'T'\n",
       "6. 'H'\n",
       "7. 'H'\n",
       "8. 'T'\n",
       "9. 'T'\n",
       "10. 'T'\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       " [1] \"T\" \"H\" \"H\" \"T\" \"T\" \"H\" \"H\" \"T\" \"T\" \"T\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t2 <- sample(c('H', 'T'), n, replace = TRUE, prob = c(0.3, 0.7))\n",
    "t2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "t2\n",
       "H T \n",
       "4 6 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "table(t2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Simulate `n` rolls of a 6-sided die"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "n <- 100"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "\t<li>6</li>\n",
       "\t<li>5</li>\n",
       "\t<li>4</li>\n",
       "\t<li>1</li>\n",
       "\t<li>4</li>\n",
       "\t<li>2</li>\n",
       "\t<li>2</li>\n",
       "\t<li>3</li>\n",
       "\t<li>5</li>\n",
       "\t<li>5</li>\n",
       "\t<li>6</li>\n",
       "\t<li>3</li>\n",
       "\t<li>6</li>\n",
       "\t<li>4</li>\n",
       "\t<li>5</li>\n",
       "\t<li>1</li>\n",
       "\t<li>4</li>\n",
       "\t<li>1</li>\n",
       "\t<li>1</li>\n",
       "\t<li>5</li>\n",
       "\t<li>6</li>\n",
       "\t<li>3</li>\n",
       "\t<li>1</li>\n",
       "\t<li>2</li>\n",
       "\t<li>5</li>\n",
       "\t<li>6</li>\n",
       "\t<li>1</li>\n",
       "\t<li>5</li>\n",
       "\t<li>6</li>\n",
       "\t<li>5</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 3\n",
       "\\item 1\n",
       "\\item 1\n",
       "\\item 3\n",
       "\\item 4\n",
       "\\item 5\n",
       "\\item 5\n",
       "\\item 5\n",
       "\\item 4\n",
       "\\item 2\n",
       "\\item 4\n",
       "\\item 3\n",
       "\\item 2\n",
       "\\item 1\n",
       "\\item 2\n",
       "\\item 6\n",
       "\\item 3\n",
       "\\item 6\n",
       "\\item 2\n",
       "\\item 4\n",
       "\\item 4\n",
       "\\item 6\n",
       "\\item 2\n",
       "\\item 5\n",
       "\\item 4\n",
       "\\item 5\n",
       "\\item 4\n",
       "\\item 2\n",
       "\\item 2\n",
       "\\item 3\n",
       "\\item 1\n",
       "\\item 5\n",
       "\\item 2\n",
       "\\item 2\n",
       "\\item 6\n",
       "\\item 6\n",
       "\\item 2\n",
       "\\item 4\n",
       "\\item 3\n",
       "\\item 6\n",
       "\\item 5\n",
       "\\item 2\n",
       "\\item 6\n",
       "\\item 2\n",
       "\\item 2\n",
       "\\item 5\n",
       "\\item 1\n",
       "\\item 1\n",
       "\\item 4\n",
       "\\item 5\n",
       "\\item 2\n",
       "\\item 1\n",
       "\\item 5\n",
       "\\item 4\n",
       "\\item 4\n",
       "\\item 1\n",
       "\\item 3\n",
       "\\item 3\n",
       "\\item 5\n",
       "\\item 5\n",
       "\\item 4\n",
       "\\item 6\n",
       "\\item 5\n",
       "\\item 4\n",
       "\\item 6\n",
       "\\item 2\n",
       "\\item 2\n",
       "\\item 1\n",
       "\\item 2\n",
       "\\item 5\n",
       "\\item 6\n",
       "\\item 5\n",
       "\\item 4\n",
       "\\item 1\n",
       "\\item 4\n",
       "\\item 2\n",
       "\\item 2\n",
       "\\item 3\n",
       "\\item 5\n",
       "\\item 5\n",
       "\\item 6\n",
       "\\item 3\n",
       "\\item 6\n",
       "\\item 4\n",
       "\\item 5\n",
       "\\item 1\n",
       "\\item 4\n",
       "\\item 1\n",
       "\\item 1\n",
       "\\item 5\n",
       "\\item 6\n",
       "\\item 3\n",
       "\\item 1\n",
       "\\item 2\n",
       "\\item 5\n",
       "\\item 6\n",
       "\\item 1\n",
       "\\item 5\n",
       "\\item 6\n",
       "\\item 5\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 3\n",
       "2. 1\n",
       "3. 1\n",
       "4. 3\n",
       "5. 4\n",
       "6. 5\n",
       "7. 5\n",
       "8. 5\n",
       "9. 4\n",
       "10. 2\n",
       "11. 4\n",
       "12. 3\n",
       "13. 2\n",
       "14. 1\n",
       "15. 2\n",
       "16. 6\n",
       "17. 3\n",
       "18. 6\n",
       "19. 2\n",
       "20. 4\n",
       "21. 4\n",
       "22. 6\n",
       "23. 2\n",
       "24. 5\n",
       "25. 4\n",
       "26. 5\n",
       "27. 4\n",
       "28. 2\n",
       "29. 2\n",
       "30. 3\n",
       "31. 1\n",
       "32. 5\n",
       "33. 2\n",
       "34. 2\n",
       "35. 6\n",
       "36. 6\n",
       "37. 2\n",
       "38. 4\n",
       "39. 3\n",
       "40. 6\n",
       "41. 5\n",
       "42. 2\n",
       "43. 6\n",
       "44. 2\n",
       "45. 2\n",
       "46. 5\n",
       "47. 1\n",
       "48. 1\n",
       "49. 4\n",
       "50. 5\n",
       "51. 2\n",
       "52. 1\n",
       "53. 5\n",
       "54. 4\n",
       "55. 4\n",
       "56. 1\n",
       "57. 3\n",
       "58. 3\n",
       "59. 5\n",
       "60. 5\n",
       "61. 4\n",
       "62. 6\n",
       "63. 5\n",
       "64. 4\n",
       "65. 6\n",
       "66. 2\n",
       "67. 2\n",
       "68. 1\n",
       "69. 2\n",
       "70. 5\n",
       "71. 6\n",
       "72. 5\n",
       "73. 4\n",
       "74. 1\n",
       "75. 4\n",
       "76. 2\n",
       "77. 2\n",
       "78. 3\n",
       "79. 5\n",
       "80. 5\n",
       "81. 6\n",
       "82. 3\n",
       "83. 6\n",
       "84. 4\n",
       "85. 5\n",
       "86. 1\n",
       "87. 4\n",
       "88. 1\n",
       "89. 1\n",
       "90. 5\n",
       "91. 6\n",
       "92. 3\n",
       "93. 1\n",
       "94. 2\n",
       "95. 5\n",
       "96. 6\n",
       "97. 1\n",
       "98. 5\n",
       "99. 6\n",
       "100. 5\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "  [1] 3 1 1 3 4 5 5 5 4 2 4 3 2 1 2 6 3 6 2 4 4 6 2 5 4 5 4 2 2 3 1 5 2 2 6 6 2\n",
       " [38] 4 3 6 5 2 6 2 2 5 1 1 4 5 2 1 5 4 4 1 3 3 5 5 4 6 5 4 6 2 2 1 2 5 6 5 4 1\n",
       " [75] 4 2 2 3 5 5 6 3 6 4 5 1 4 1 1 5 6 3 1 2 5 6 1 5 6 5"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "d1 <- sample(1:6, n, replace=TRUE)\n",
    "d1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "d1\n",
       " 1  2  3  4  5  6 \n",
       "15 20 11 17 22 15 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "table(d1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sampling without replacement. For example, if we wanted to assiggn 16 samples to treatment A or B at random such that exactly half had each treatment."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>'B'</li>\n",
       "\t<li>'A'</li>\n",
       "\t<li>'B'</li>\n",
       "\t<li>'B'</li>\n",
       "\t<li>'B'</li>\n",
       "\t<li>'B'</li>\n",
       "\t<li>'A'</li>\n",
       "\t<li>'A'</li>\n",
       "\t<li>'A'</li>\n",
       "\t<li>'B'</li>\n",
       "\t<li>'A'</li>\n",
       "\t<li>'B'</li>\n",
       "\t<li>'B'</li>\n",
       "\t<li>'A'</li>\n",
       "\t<li>'A'</li>\n",
       "\t<li>'A'</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 'B'\n",
       "\\item 'A'\n",
       "\\item 'B'\n",
       "\\item 'B'\n",
       "\\item 'B'\n",
       "\\item 'B'\n",
       "\\item 'A'\n",
       "\\item 'A'\n",
       "\\item 'A'\n",
       "\\item 'B'\n",
       "\\item 'A'\n",
       "\\item 'B'\n",
       "\\item 'B'\n",
       "\\item 'A'\n",
       "\\item 'A'\n",
       "\\item 'A'\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 'B'\n",
       "2. 'A'\n",
       "3. 'B'\n",
       "4. 'B'\n",
       "5. 'B'\n",
       "6. 'B'\n",
       "7. 'A'\n",
       "8. 'A'\n",
       "9. 'A'\n",
       "10. 'B'\n",
       "11. 'A'\n",
       "12. 'B'\n",
       "13. 'B'\n",
       "14. 'A'\n",
       "15. 'A'\n",
       "16. 'A'\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       " [1] \"B\" \"A\" \"B\" \"B\" \"B\" \"B\" \"A\" \"A\" \"A\" \"B\" \"A\" \"B\" \"B\" \"A\" \"A\" \"A\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sample(rep(c('A', 'B'), each=8))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Random number generators"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Discrete distributions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sampling from a Bernoullli distribution returns TRUE for success and FALSE for failure."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>TRUE</li>\n",
       "\t<li>TRUE</li>\n",
       "\t<li>FALSE</li>\n",
       "\t<li>TRUE</li>\n",
       "\t<li>FALSE</li>\n",
       "\t<li>FALSE</li>\n",
       "\t<li>FALSE</li>\n",
       "\t<li>FALSE</li>\n",
       "\t<li>FALSE</li>\n",
       "\t<li>TRUE</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item TRUE\n",
       "\\item TRUE\n",
       "\\item FALSE\n",
       "\\item TRUE\n",
       "\\item FALSE\n",
       "\\item FALSE\n",
       "\\item FALSE\n",
       "\\item FALSE\n",
       "\\item FALSE\n",
       "\\item TRUE\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. TRUE\n",
       "2. TRUE\n",
       "3. FALSE\n",
       "4. TRUE\n",
       "5. FALSE\n",
       "6. FALSE\n",
       "7. FALSE\n",
       "8. FALSE\n",
       "9. FALSE\n",
       "10. TRUE\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       " [1]  TRUE  TRUE FALSE  TRUE FALSE FALSE FALSE FALSE FALSE  TRUE"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rbernoulli(n=10, p=0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>0</li>\n",
       "\t<li>0</li>\n",
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       "\t<li>1</li>\n",
       "\t<li>1</li>\n",
       "\t<li>1</li>\n",
       "</ol>\n"
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      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 0\n",
       "\\item 0\n",
       "\\item 0\n",
       "\\item 0\n",
       "\\item 0\n",
       "\\item 1\n",
       "\\item 1\n",
       "\\item 1\n",
       "\\item 1\n",
       "\\item 1\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 0\n",
       "2. 0\n",
       "3. 0\n",
       "4. 0\n",
       "5. 0\n",
       "6. 1\n",
       "7. 1\n",
       "8. 1\n",
       "9. 1\n",
       "10. 1\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       " [1] 0 0 0 0 0 1 1 1 1 1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "as.integer(rbernoulli(n=10, p=0.5))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sampling from a Binomial distribution returns the number of **successes** in `size` trials for `n` experiments."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>4</li>\n",
       "\t<li>2</li>\n",
       "\t<li>2</li>\n",
       "\t<li>3</li>\n",
       "\t<li>3</li>\n",
       "\t<li>4</li>\n",
       "\t<li>3</li>\n",
       "\t<li>1</li>\n",
       "\t<li>2</li>\n",
       "\t<li>2</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 4\n",
       "\\item 2\n",
       "\\item 2\n",
       "\\item 3\n",
       "\\item 3\n",
       "\\item 4\n",
       "\\item 3\n",
       "\\item 1\n",
       "\\item 2\n",
       "\\item 2\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 4\n",
       "2. 2\n",
       "3. 2\n",
       "4. 3\n",
       "5. 3\n",
       "6. 4\n",
       "7. 3\n",
       "8. 1\n",
       "9. 2\n",
       "10. 2\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       " [1] 4 2 2 3 3 4 3 1 2 2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rbinom(n=10, p=0.5, size=5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sampling from a negative binomial distribution returns the number of **failures** until `size` succcesses are observed for `n` experiemnts."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>4</li>\n",
       "\t<li>0</li>\n",
       "\t<li>9</li>\n",
       "\t<li>4</li>\n",
       "\t<li>2</li>\n",
       "\t<li>4</li>\n",
       "\t<li>6</li>\n",
       "\t<li>5</li>\n",
       "\t<li>5</li>\n",
       "\t<li>4</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 4\n",
       "\\item 0\n",
       "\\item 9\n",
       "\\item 4\n",
       "\\item 2\n",
       "\\item 4\n",
       "\\item 6\n",
       "\\item 5\n",
       "\\item 5\n",
       "\\item 4\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 4\n",
       "2. 0\n",
       "3. 9\n",
       "4. 4\n",
       "5. 2\n",
       "6. 4\n",
       "7. 6\n",
       "8. 5\n",
       "9. 5\n",
       "10. 4\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       " [1] 4 0 9 4 2 4 6 5 5 4"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rnbinom(n=10, size=5, prob=0.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sampling from a Poisson distribution returns the number of **successes** in `n` experiments if the average success rate per experiment is `lambda`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>5</li>\n",
       "\t<li>2</li>\n",
       "\t<li>1</li>\n",
       "\t<li>0</li>\n",
       "\t<li>2</li>\n",
       "\t<li>2</li>\n",
       "\t<li>1</li>\n",
       "\t<li>2</li>\n",
       "\t<li>0</li>\n",
       "\t<li>9</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 5\n",
       "\\item 2\n",
       "\\item 1\n",
       "\\item 0\n",
       "\\item 2\n",
       "\\item 2\n",
       "\\item 1\n",
       "\\item 2\n",
       "\\item 0\n",
       "\\item 9\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 5\n",
       "2. 2\n",
       "3. 1\n",
       "4. 0\n",
       "5. 2\n",
       "6. 2\n",
       "7. 1\n",
       "8. 2\n",
       "9. 0\n",
       "10. 9\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       " [1] 5 2 1 0 2 2 1 2 0 9"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rpois(n=10, lambda = 3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Note**:  We can give different parameters for each experiment in these distributions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>2</li>\n",
       "\t<li>0</li>\n",
       "\t<li>5</li>\n",
       "\t<li>4</li>\n",
       "\t<li>4</li>\n",
       "\t<li>3</li>\n",
       "\t<li>7</li>\n",
       "\t<li>4</li>\n",
       "\t<li>7</li>\n",
       "\t<li>10</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 2\n",
       "\\item 0\n",
       "\\item 5\n",
       "\\item 4\n",
       "\\item 4\n",
       "\\item 3\n",
       "\\item 7\n",
       "\\item 4\n",
       "\\item 7\n",
       "\\item 10\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 2\n",
       "2. 0\n",
       "3. 5\n",
       "4. 4\n",
       "5. 4\n",
       "6. 3\n",
       "7. 7\n",
       "8. 4\n",
       "9. 7\n",
       "10. 10\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       " [1]  2  0  5  4  4  3  7  4  7 10"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rpois(n=10, lambda=1:10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Continuous distributions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sampling from a standard uniform distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>0.159469854552299</li>\n",
       "\t<li>0.149578995071352</li>\n",
       "\t<li>0.499272880377248</li>\n",
       "\t<li>0.940564878052101</li>\n",
       "\t<li>0.334231326589361</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 0.159469854552299\n",
       "\\item 0.149578995071352\n",
       "\\item 0.499272880377248\n",
       "\\item 0.940564878052101\n",
       "\\item 0.334231326589361\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 0.159469854552299\n",
       "2. 0.149578995071352\n",
       "3. 0.499272880377248\n",
       "4. 0.940564878052101\n",
       "5. 0.334231326589361\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 0.1594699 0.1495790 0.4992729 0.9405649 0.3342313"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "runif(5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sampling form a uniform distribution, with values between 90 and 100."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>91.8843432958238</li>\n",
       "\t<li>92.6971617829986</li>\n",
       "\t<li>95.3074407810345</li>\n",
       "\t<li>90.2145022852346</li>\n",
       "\t<li>97.987603074871</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 91.8843432958238\n",
       "\\item 92.6971617829986\n",
       "\\item 95.3074407810345\n",
       "\\item 90.2145022852346\n",
       "\\item 97.987603074871\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 91.8843432958238\n",
       "2. 92.6971617829986\n",
       "3. 95.3074407810345\n",
       "4. 90.2145022852346\n",
       "5. 97.987603074871\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 91.88434 92.69716 95.30744 90.21450 97.98760"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "runif(5, 90, 100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sampling from a standard normal distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>-1.22474795035999</li>\n",
       "\t<li>0.179516441117938</li>\n",
       "\t<li>0.567620594423535</li>\n",
       "\t<li>-0.492877353553475</li>\n",
       "\t<li>6.28840653511241e-05</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item -1.22474795035999\n",
       "\\item 0.179516441117938\n",
       "\\item 0.567620594423535\n",
       "\\item -0.492877353553475\n",
       "\\item 6.28840653511241e-05\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. -1.22474795035999\n",
       "2. 0.179516441117938\n",
       "3. 0.567620594423535\n",
       "4. -0.492877353553475\n",
       "5. 6.28840653511241e-05\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] -1.224748e+00  1.795164e-01  5.676206e-01 -4.928774e-01  6.288407e-05"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rnorm(5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Looping"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 0.06091411\n",
      "[1] -0.1028669\n",
      "[1] -0.3117706\n",
      "[1] 0.09639036\n",
      "[1] 0.2010454\n",
      "[1] -0.2384309\n",
      "[1] 0.1271087\n",
      "[1] -0.2115652\n",
      "[1] -0.01946474\n",
      "[1] -0.04661421\n"
     ]
    }
   ],
   "source": [
    "for (i in 1:10) {\n",
    "    print(mean(rnorm(10)))\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Saving variables generated in a loop"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>0.102869685091412</li>\n",
       "\t<li>0.0394963601049801</li>\n",
       "\t<li>0.0587241371393688</li>\n",
       "\t<li>-0.208039597462606</li>\n",
       "\t<li>0.175088960315561</li>\n",
       "\t<li>-0.0224984001571429</li>\n",
       "\t<li>0.157365195356717</li>\n",
       "\t<li>-0.11380999366832</li>\n",
       "\t<li>-0.151701911199645</li>\n",
       "\t<li>0.0806058319328881</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 0.102869685091412\n",
       "\\item 0.0394963601049801\n",
       "\\item 0.0587241371393688\n",
       "\\item -0.208039597462606\n",
       "\\item 0.175088960315561\n",
       "\\item -0.0224984001571429\n",
       "\\item 0.157365195356717\n",
       "\\item -0.11380999366832\n",
       "\\item -0.151701911199645\n",
       "\\item 0.0806058319328881\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 0.102869685091412\n",
       "2. 0.0394963601049801\n",
       "3. 0.0587241371393688\n",
       "4. -0.208039597462606\n",
       "5. 0.175088960315561\n",
       "6. -0.0224984001571429\n",
       "7. 0.157365195356717\n",
       "8. -0.11380999366832\n",
       "9. -0.151701911199645\n",
       "10. 0.0806058319328881\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       " [1]  0.10286969  0.03949636  0.05872414 -0.20803960  0.17508896 -0.02249840\n",
       " [7]  0.15736520 -0.11380999 -0.15170191  0.08060583"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n <- 10\n",
    "vars <- numeric(n)\n",
    "for (i in 1:n) {\n",
    "    vars[i] <- mean(rnorm(10))\n",
    "}\n",
    "vars"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "\t<li>0.152764106747895</li>\n",
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       "\t<li><ol class=list-inline>\n",
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       "\t<li><ol class=list-inline>\n",
       "\t<li>-0.527108125374458</li>\n",
       "\t<li>-0.0214270650416246</li>\n",
       "\t<li>0.670498070996282</li>\n",
       "\t<li>-0.434617038572407</li>\n",
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       "</li>\n",
       "\t<li><ol class=list-inline>\n",
       "\t<li>-1.11387978331285</li>\n",
       "\t<li>0.6071059948785</li>\n",
       "\t<li>0.275456968739591</li>\n",
       "\t<li>1.15734706941474</li>\n",
       "\t<li>-1.68248085947826</li>\n",
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       "\t<li><ol class=list-inline>\n",
       "\t<li>0.0873190888948744</li>\n",
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       "\t<li>0.724173800683369</li>\n",
       "\t<li>-0.832552825764212</li>\n",
       "\t<li>0.732528486768414</li>\n",
       "\t<li>-0.871926869997213</li>\n",
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       "</li>\n",
       "\t<li><ol class=list-inline>\n",
       "\t<li>-0.453397511229793</li>\n",
       "\t<li>1.18753427862908</li>\n",
       "\t<li>-0.290145311790079</li>\n",
       "\t<li>0.828546144965436</li>\n",
       "\t<li>-0.291227708779169</li>\n",
       "\t<li>-1.57636240474791</li>\n",
       "\t<li>-0.848815696948922</li>\n",
       "</ol>\n",
       "</li>\n",
       "\t<li><ol class=list-inline>\n",
       "\t<li>-1.0885198619572</li>\n",
       "\t<li>-0.484290571117173</li>\n",
       "\t<li>-0.336311209009573</li>\n",
       "\t<li>-0.153357890661644</li>\n",
       "\t<li>-0.243247228574846</li>\n",
       "\t<li>1.89220204160011</li>\n",
       "\t<li>-1.38599833728542</li>\n",
       "\t<li>-0.41482430070778</li>\n",
       "</ol>\n",
       "</li>\n",
       "\t<li><ol class=list-inline>\n",
       "\t<li>0.349081528106831</li>\n",
       "\t<li>1.62844226581541</li>\n",
       "\t<li>0.0885218957136163</li>\n",
       "\t<li>1.23915070840056</li>\n",
       "\t<li>-1.64455553575371</li>\n",
       "\t<li>1.44635652537848</li>\n",
       "\t<li>-0.690560171665183</li>\n",
       "\t<li>-0.276431085453852</li>\n",
       "\t<li>-1.10941875989944</li>\n",
       "</ol>\n",
       "</li>\n",
       "\t<li><ol class=list-inline>\n",
       "\t<li>0.133869316429452</li>\n",
       "\t<li>1.78533905173155</li>\n",
       "\t<li>2.42216335525159</li>\n",
       "\t<li>-1.07682890211841</li>\n",
       "\t<li>0.485941110403099</li>\n",
       "\t<li>1.38852173874429</li>\n",
       "\t<li>-0.195656817284544</li>\n",
       "\t<li>-0.218174797706211</li>\n",
       "\t<li>-0.304777954546445</li>\n",
       "\t<li>0.597832724111443</li>\n",
       "</ol>\n",
       "</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate}\n",
       "\\item 0.152764106747895\n",
       "\\item \\begin{enumerate*}\n",
       "\\item 0.988596845178799\n",
       "\\item -0.073458334672174\n",
       "\\end{enumerate*}\n",
       "\n",
       "\\item \\begin{enumerate*}\n",
       "\\item -1.38702655355974\n",
       "\\item -1.30667590437536\n",
       "\\item -0.768395325077939\n",
       "\\end{enumerate*}\n",
       "\n",
       "\\item \\begin{enumerate*}\n",
       "\\item -0.527108125374458\n",
       "\\item -0.0214270650416246\n",
       "\\item 0.670498070996282\n",
       "\\item -0.434617038572407\n",
       "\\end{enumerate*}\n",
       "\n",
       "\\item \\begin{enumerate*}\n",
       "\\item -1.11387978331285\n",
       "\\item 0.6071059948785\n",
       "\\item 0.275456968739591\n",
       "\\item 1.15734706941474\n",
       "\\item -1.68248085947826\n",
       "\\end{enumerate*}\n",
       "\n",
       "\\item \\begin{enumerate*}\n",
       "\\item 0.0873190888948744\n",
       "\\item 1.35336189394267\n",
       "\\item 0.724173800683369\n",
       "\\item -0.832552825764212\n",
       "\\item 0.732528486768414\n",
       "\\item -0.871926869997213\n",
       "\\end{enumerate*}\n",
       "\n",
       "\\item \\begin{enumerate*}\n",
       "\\item -0.453397511229793\n",
       "\\item 1.18753427862908\n",
       "\\item -0.290145311790079\n",
       "\\item 0.828546144965436\n",
       "\\item -0.291227708779169\n",
       "\\item -1.57636240474791\n",
       "\\item -0.848815696948922\n",
       "\\end{enumerate*}\n",
       "\n",
       "\\item \\begin{enumerate*}\n",
       "\\item -1.0885198619572\n",
       "\\item -0.484290571117173\n",
       "\\item -0.336311209009573\n",
       "\\item -0.153357890661644\n",
       "\\item -0.243247228574846\n",
       "\\item 1.89220204160011\n",
       "\\item -1.38599833728542\n",
       "\\item -0.41482430070778\n",
       "\\end{enumerate*}\n",
       "\n",
       "\\item \\begin{enumerate*}\n",
       "\\item 0.349081528106831\n",
       "\\item 1.62844226581541\n",
       "\\item 0.0885218957136163\n",
       "\\item 1.23915070840056\n",
       "\\item -1.64455553575371\n",
       "\\item 1.44635652537848\n",
       "\\item -0.690560171665183\n",
       "\\item -0.276431085453852\n",
       "\\item -1.10941875989944\n",
       "\\end{enumerate*}\n",
       "\n",
       "\\item \\begin{enumerate*}\n",
       "\\item 0.133869316429452\n",
       "\\item 1.78533905173155\n",
       "\\item 2.42216335525159\n",
       "\\item -1.07682890211841\n",
       "\\item 0.485941110403099\n",
       "\\item 1.38852173874429\n",
       "\\item -0.195656817284544\n",
       "\\item -0.218174797706211\n",
       "\\item -0.304777954546445\n",
       "\\item 0.597832724111443\n",
       "\\end{enumerate*}\n",
       "\n",
       "\\end{enumerate}\n"
      ],
      "text/markdown": [
       "1. 0.152764106747895\n",
       "2. 1. 0.988596845178799\n",
       "2. -0.073458334672174\n",
       "\n",
       "\n",
       "\n",
       "3. 1. -1.38702655355974\n",
       "2. -1.30667590437536\n",
       "3. -0.768395325077939\n",
       "\n",
       "\n",
       "\n",
       "4. 1. -0.527108125374458\n",
       "2. -0.0214270650416246\n",
       "3. 0.670498070996282\n",
       "4. -0.434617038572407\n",
       "\n",
       "\n",
       "\n",
       "5. 1. -1.11387978331285\n",
       "2. 0.6071059948785\n",
       "3. 0.275456968739591\n",
       "4. 1.15734706941474\n",
       "5. -1.68248085947826\n",
       "\n",
       "\n",
       "\n",
       "6. 1. 0.0873190888948744\n",
       "2. 1.35336189394267\n",
       "3. 0.724173800683369\n",
       "4. -0.832552825764212\n",
       "5. 0.732528486768414\n",
       "6. -0.871926869997213\n",
       "\n",
       "\n",
       "\n",
       "7. 1. -0.453397511229793\n",
       "2. 1.18753427862908\n",
       "3. -0.290145311790079\n",
       "4. 0.828546144965436\n",
       "5. -0.291227708779169\n",
       "6. -1.57636240474791\n",
       "7. -0.848815696948922\n",
       "\n",
       "\n",
       "\n",
       "8. 1. -1.0885198619572\n",
       "2. -0.484290571117173\n",
       "3. -0.336311209009573\n",
       "4. -0.153357890661644\n",
       "5. -0.243247228574846\n",
       "6. 1.89220204160011\n",
       "7. -1.38599833728542\n",
       "8. -0.41482430070778\n",
       "\n",
       "\n",
       "\n",
       "9. 1. 0.349081528106831\n",
       "2. 1.62844226581541\n",
       "3. 0.0885218957136163\n",
       "4. 1.23915070840056\n",
       "5. -1.64455553575371\n",
       "6. 1.44635652537848\n",
       "7. -0.690560171665183\n",
       "8. -0.276431085453852\n",
       "9. -1.10941875989944\n",
       "\n",
       "\n",
       "\n",
       "10. 1. 0.133869316429452\n",
       "2. 1.78533905173155\n",
       "3. 2.42216335525159\n",
       "4. -1.07682890211841\n",
       "5. 0.485941110403099\n",
       "6. 1.38852173874429\n",
       "7. -0.195656817284544\n",
       "8. -0.218174797706211\n",
       "9. -0.304777954546445\n",
       "10. 0.597832724111443\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[[1]]\n",
       "[1] 0.1527641\n",
       "\n",
       "[[2]]\n",
       "[1]  0.98859685 -0.07345833\n",
       "\n",
       "[[3]]\n",
       "[1] -1.3870266 -1.3066759 -0.7683953\n",
       "\n",
       "[[4]]\n",
       "[1] -0.52710813 -0.02142707  0.67049807 -0.43461704\n",
       "\n",
       "[[5]]\n",
       "[1] -1.113880  0.607106  0.275457  1.157347 -1.682481\n",
       "\n",
       "[[6]]\n",
       "[1]  0.08731909  1.35336189  0.72417380 -0.83255283  0.73252849 -0.87192687\n",
       "\n",
       "[[7]]\n",
       "[1] -0.4533975  1.1875343 -0.2901453  0.8285461 -0.2912277 -1.5763624 -0.8488157\n",
       "\n",
       "[[8]]\n",
       "[1] -1.0885199 -0.4842906 -0.3363112 -0.1533579 -0.2432472  1.8922020 -1.3859983\n",
       "[8] -0.4148243\n",
       "\n",
       "[[9]]\n",
       "[1]  0.3490815  1.6284423  0.0885219  1.2391507 -1.6445555  1.4463565 -0.6905602\n",
       "[8] -0.2764311 -1.1094188\n",
       "\n",
       "[[10]]\n",
       " [1]  0.1338693  1.7853391  2.4221634 -1.0768289  0.4859411  1.3885217\n",
       " [7] -0.1956568 -0.2181748 -0.3047780  0.5978327\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "lapply(1:10,FUN = rnorm)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol>\n",
       "\t<li>1.39742941081576</li>\n",
       "\t<li>0.503903887768466</li>\n",
       "\t<li>-0.117686052712023</li>\n",
       "\t<li>0.375071580419853</li>\n",
       "\t<li>0.239310604458899</li>\n",
       "\t<li>0.100315651769656</li>\n",
       "\t<li>-0.976662058729188</li>\n",
       "\t<li>-0.279974621278527</li>\n",
       "\t<li>0.203381481503197</li>\n",
       "\t<li>0.0364462992139071</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate}\n",
       "\\item 1.39742941081576\n",
       "\\item 0.503903887768466\n",
       "\\item -0.117686052712023\n",
       "\\item 0.375071580419853\n",
       "\\item 0.239310604458899\n",
       "\\item 0.100315651769656\n",
       "\\item -0.976662058729188\n",
       "\\item -0.279974621278527\n",
       "\\item 0.203381481503197\n",
       "\\item 0.0364462992139071\n",
       "\\end{enumerate}\n"
      ],
      "text/markdown": [
       "1. 1.39742941081576\n",
       "2. 0.503903887768466\n",
       "3. -0.117686052712023\n",
       "4. 0.375071580419853\n",
       "5. 0.239310604458899\n",
       "6. 0.100315651769656\n",
       "7. -0.976662058729188\n",
       "8. -0.279974621278527\n",
       "9. 0.203381481503197\n",
       "10. 0.0364462992139071\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[[1]]\n",
       "[1] 1.397429\n",
       "\n",
       "[[2]]\n",
       "[1] 0.5039039\n",
       "\n",
       "[[3]]\n",
       "[1] -0.1176861\n",
       "\n",
       "[[4]]\n",
       "[1] 0.3750716\n",
       "\n",
       "[[5]]\n",
       "[1] 0.2393106\n",
       "\n",
       "[[6]]\n",
       "[1] 0.1003157\n",
       "\n",
       "[[7]]\n",
       "[1] -0.9766621\n",
       "\n",
       "[[8]]\n",
       "[1] -0.2799746\n",
       "\n",
       "[[9]]\n",
       "[1] 0.2033815\n",
       "\n",
       "[[10]]\n",
       "[1] 0.0364463\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "lapply(1:10, FUN = function(x) mean(rnorm(x)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Using `replicate`\n",
    "\n",
    "`replicate` is like `rep` but works for a function (such as a random number generator)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<caption>A matrix: 5 × 3 of type dbl</caption>\n",
       "<tbody>\n",
       "\t<tr><td>-0.8673179</td><td>-0.2785431</td><td>-0.4034675</td></tr>\n",
       "\t<tr><td> 0.9506517</td><td> 0.5461152</td><td> 0.1046594</td></tr>\n",
       "\t<tr><td>-0.5850115</td><td>-1.3038212</td><td>-0.3188808</td></tr>\n",
       "\t<tr><td> 0.3209575</td><td>-0.2509145</td><td> 1.6183439</td></tr>\n",
       "\t<tr><td>-0.2993960</td><td> 0.1710074</td><td> 0.7141886</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A matrix: 5 × 3 of type dbl\n",
       "\\begin{tabular}{lll}\n",
       "\t -0.8673179 & -0.2785431 & -0.4034675\\\\\n",
       "\t  0.9506517 &  0.5461152 &  0.1046594\\\\\n",
       "\t -0.5850115 & -1.3038212 & -0.3188808\\\\\n",
       "\t  0.3209575 & -0.2509145 &  1.6183439\\\\\n",
       "\t -0.2993960 &  0.1710074 &  0.7141886\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A matrix: 5 × 3 of type dbl\n",
       "\n",
       "| -0.8673179 | -0.2785431 | -0.4034675 |\n",
       "|  0.9506517 |  0.5461152 |  0.1046594 |\n",
       "| -0.5850115 | -1.3038212 | -0.3188808 |\n",
       "|  0.3209575 | -0.2509145 |  1.6183439 |\n",
       "| -0.2993960 |  0.1710074 |  0.7141886 |\n",
       "\n"
      ],
      "text/plain": [
       "     [,1]       [,2]       [,3]      \n",
       "[1,] -0.8673179 -0.2785431 -0.4034675\n",
       "[2,]  0.9506517  0.5461152  0.1046594\n",
       "[3,] -0.5850115 -1.3038212 -0.3188808\n",
       "[4,]  0.3209575 -0.2509145  1.6183439\n",
       "[5,] -0.2993960  0.1710074  0.7141886"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "replicate(3, rnorm(5))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`replicate` is quite useful for simulations. For example, suppose we want to know the distribution of the sample mean  if we sampled 100 numbers from the standard normal distribution 1,000 times."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [],
   "source": [
    "n_expts <- 1000\n",
    "n <- 100"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Using for loop"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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LMjacYTs+AHbvdOg3RlxCwLNdl3VY5N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\nA6e7+7xSj5WPAAAAAElFTkSuQmCC",
      "text/plain": [
       "Plot with title “Histogram of mus”"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "set.seed(123)\n",
    "mus <- numeric(n_expts)\n",
    "for (i in 1:n_expts) {\n",
    "    mus[i] <- mean(rnorm(n))\n",
    "}\n",
    "hist(mus)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Using replicate"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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LMjacYTs+AHbvdOg3RlxCwLNdl3VY5N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\nA6e7+7xSj5WPAAAAAElFTkSuQmCC",
      "text/plain": [
       "Plot with title “Histogram of mus”"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "set.seed(123)\n",
    "mus <- replicate(n_expts, mean(rnorm(n)))\n",
    "hist(mus)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Making a `data.frame`  or `tibble` of simulated data\n",
    "\n",
    "Let's simulate the following experiment. \n",
    "\n",
    "- There are 10 subjects in Group A and 10 subjects in Group B with random PIDs from 10000-99999\n",
    "- We measure 5 genes in each subject. The genes have the same distribution for each subject, but different genes have different distribtutions:\n",
    "    - gene1 $\\sim N(10, 1)$\n",
    "    - gene2 $\\sim N(11, 2)$\n",
    "    - gene3 $\\sim N(12, 3)$\n",
    "    - gene4 $\\sim N(13, 4)$\n",
    "    - gene5 $\\sim (N(14, 5)$\n",
    "    \n",
    "NB: This is for illustration purposes! Our gene counts (even if 'normalized') are not normally distributed!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<caption>A matrix: 3 × 5 of type dbl</caption>\n",
       "<tbody>\n",
       "\t<tr><td>1.264993</td><td>0.9467906</td><td>1.076689</td><td>-1.503406</td><td>1.069951</td></tr>\n",
       "\t<tr><td>3.830747</td><td>2.4379042</td><td>2.455073</td><td> 1.404136</td><td>3.169101</td></tr>\n",
       "\t<tr><td>2.940622</td><td>4.3374490</td><td>3.873871</td><td> 2.264472</td><td>1.632268</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A matrix: 3 × 5 of type dbl\n",
       "\\begin{tabular}{lllll}\n",
       "\t 1.264993 & 0.9467906 & 1.076689 & -1.503406 & 1.069951\\\\\n",
       "\t 3.830747 & 2.4379042 & 2.455073 &  1.404136 & 3.169101\\\\\n",
       "\t 2.940622 & 4.3374490 & 3.873871 &  2.264472 & 1.632268\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A matrix: 3 × 5 of type dbl\n",
       "\n",
       "| 1.264993 | 0.9467906 | 1.076689 | -1.503406 | 1.069951 |\n",
       "| 3.830747 | 2.4379042 | 2.455073 |  1.404136 | 3.169101 |\n",
       "| 2.940622 | 4.3374490 | 3.873871 |  2.264472 | 1.632268 |\n",
       "\n"
      ],
      "text/plain": [
       "     [,1]     [,2]      [,3]     [,4]      [,5]    \n",
       "[1,] 1.264993 0.9467906 1.076689 -1.503406 1.069951\n",
       "[2,] 3.830747 2.4379042 2.455073  1.404136 3.169101\n",
       "[3,] 2.940622 4.3374490 3.873871  2.264472 1.632268"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "replicate(5, rnorm(3, 1:5, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [],
   "source": [
    "n <- 10\n",
    "n_genes <- 5\n",
    "min_pid <- 10000\n",
    "max_pid <- 99999\n",
    "groupings <- c('A', 'B')\n",
    "n_groups <- length(groupings)\n",
    "gene_mus <- 10:14\n",
    "gene_sigmas <- 1:5\n",
    "pad_width <- 3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [],
   "source": [
    "pids <- sample(min_pid:max_pid, n_groups*n)\n",
    "groups <- sample(rep(groupings, each=n))\n",
    "genes <- t(replicate(n_groups*n, rnorm(n_genes, gene_mus, gene_sigmas)))\n",
    "gene_names <- paste('gene', str_pad(1:n_genes, width = pad_width, pad='0'), sep='')\n",
    "colnames(genes) <- gene_names"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [],
   "source": [
    "df <- tibble(\n",
    "    pid = pids,\n",
    "    grp = groups,\n",
    "    genes\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<caption>A tibble: 3 × 3</caption>\n",
       "<thead>\n",
       "\t<tr><th scope=col>pid</th><th scope=col>grp</th><th scope=col>genes</th></tr>\n",
       "\t<tr><th scope=col>&lt;int&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;dbl[,5]&gt;</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><td>97320</td><td>A</td><td> 9.532829, 11.047354, 14.08499, 14.35483, 22.422266</td></tr>\n",
       "\t<tr><td>50148</td><td>B</td><td>10.046358, 11.926989, 14.43202, 14.20053, 14.613053</td></tr>\n",
       "\t<tr><td>86850</td><td>A</td><td> 9.535037,  9.418596, 13.01083, 13.11000,  9.554716</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A tibble: 3 × 3\n",
       "\\begin{tabular}{r|lll}\n",
       " pid & grp & genes\\\\\n",
       " <int> & <chr> & <dbl{[},5{]}>\\\\\n",
       "\\hline\n",
       "\t 97320 & A &  9.532829, 11.047354, 14.08499, 14.35483, 22.422266\\\\\n",
       "\t 50148 & B & 10.046358, 11.926989, 14.43202, 14.20053, 14.613053\\\\\n",
       "\t 86850 & A &  9.535037,  9.418596, 13.01083, 13.11000,  9.554716\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A tibble: 3 × 3\n",
       "\n",
       "| pid &lt;int&gt; | grp &lt;chr&gt; | genes &lt;dbl[,5]&gt; |\n",
       "|---|---|---|\n",
       "| 97320 | A |  9.532829, 11.047354, 14.08499, 14.35483, 22.422266 |\n",
       "| 50148 | B | 10.046358, 11.926989, 14.43202, 14.20053, 14.613053 |\n",
       "| 86850 | A |  9.535037,  9.418596, 13.01083, 13.11000,  9.554716 |\n",
       "\n"
      ],
      "text/plain": [
       "  pid   grp genes                                              \n",
       "1 97320 A    9.532829, 11.047354, 14.08499, 14.35483, 22.422266\n",
       "2 50148 B   10.046358, 11.926989, 14.43202, 14.20053, 14.613053\n",
       "3 86850 A    9.535037,  9.418596, 13.01083, 13.11000,  9.554716"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sample_n(df, 3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Breakdown of simulation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Set up simulation configration parameters."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [],
   "source": [
    "n <- 10\n",
    "n_genes <- 5\n",
    "min_pid <- 10000\n",
    "max_pid <- 99999\n",
    "groupings <- c('A', 'B')\n",
    "n_groups <- length(groupings)\n",
    "gene_mus <- 10:14\n",
    "gene_sigmas <- 1:5\n",
    "pad_width <- 3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create unique PIDs for each subject"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [],
   "source": [
    "pids <- sample(min_pid:max_pid, n_groups*n)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Assign a group to each subject at random"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [],
   "source": [
    "groups <- sample(rep(groupings, each=n))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Make up 5 genes from different distributions for each subject"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [],
   "source": [
    "genes <- t(replicate(n_groups*n, rnorm(n_genes, gene_mus, gene_sigmas)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Make nice names for each gene"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [],
   "source": [
    "gene_names <- paste('gene', str_pad(1:n_genes, width = pad_width, pad='0'), sep='')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Assign names to gene columns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [],
   "source": [
    "colnames(genes) <- gene_names"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create `tibble` to store simulated data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [],
   "source": [
    "df <- tibble(\n",
    "    pid = pids,\n",
    "    grp = groups,\n",
    "    genes\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Peek into `tibble`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<caption>A tibble: 3 × 3</caption>\n",
       "<thead>\n",
       "\t<tr><th scope=col>pid</th><th scope=col>grp</th><th scope=col>genes</th></tr>\n",
       "\t<tr><th scope=col>&lt;int&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;dbl[,5]&gt;</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><td>48611</td><td>A</td><td>11.118810, 10.281882,  9.460913, 21.654971, 12.769903</td></tr>\n",
       "\t<tr><td>82193</td><td>B</td><td> 9.250903, 12.410494, 10.502977,  8.569185, 15.181737</td></tr>\n",
       "\t<tr><td>66967</td><td>A</td><td>11.680328,  9.572095, 13.354410,  9.423921,  6.070365</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A tibble: 3 × 3\n",
       "\\begin{tabular}{r|lll}\n",
       " pid & grp & genes\\\\\n",
       " <int> & <chr> & <dbl{[},5{]}>\\\\\n",
       "\\hline\n",
       "\t 48611 & A & 11.118810, 10.281882,  9.460913, 21.654971, 12.769903\\\\\n",
       "\t 82193 & B &  9.250903, 12.410494, 10.502977,  8.569185, 15.181737\\\\\n",
       "\t 66967 & A & 11.680328,  9.572095, 13.354410,  9.423921,  6.070365\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A tibble: 3 × 3\n",
       "\n",
       "| pid &lt;int&gt; | grp &lt;chr&gt; | genes &lt;dbl[,5]&gt; |\n",
       "|---|---|---|\n",
       "| 48611 | A | 11.118810, 10.281882,  9.460913, 21.654971, 12.769903 |\n",
       "| 82193 | B |  9.250903, 12.410494, 10.502977,  8.569185, 15.181737 |\n",
       "| 66967 | A | 11.680328,  9.572095, 13.354410,  9.423921,  6.070365 |\n",
       "\n"
      ],
      "text/plain": [
       "  pid   grp genes                                                \n",
       "1 48611 A   11.118810, 10.281882,  9.460913, 21.654971, 12.769903\n",
       "2 82193 B    9.250903, 12.410494, 10.502977,  8.569185, 15.181737\n",
       "3 66967 A   11.680328,  9.572095, 13.354410,  9.423921,  6.070365"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sample_n(df, 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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