{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Count Models"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Warning message:\n",
      "“Installed Rcpp (0.12.12) different from Rcpp used to build dplyr (0.12.11).\n",
      "Please reinstall dplyr to avoid random crashes or undefined behavior.”Warning message:\n",
      "“package ‘dplyr’ was built under R version 3.4.1”"
     ]
    }
   ],
   "source": [
    "suppressPackageStartupMessages(library(tidyverse))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Coin toss model\n",
    "\n",
    "We model Heads as success with value 1 and probability $p$. Conversely, we model Tails as failure with value 0 with probability $1-p$. Suppose we toss 3 coins - these are the possible outcomes:\n",
    "\n",
    "```\n",
    "000\n",
    "001\n",
    "010\n",
    "011\n",
    "100\n",
    "101\n",
    "110\n",
    "111\n",
    "```\n",
    "\n",
    "It is easy to see that there are $2^k$ possible outcomes when we toss $k$ coins (or equivlaently toss one coin $k$ times). In statistics, each \"toss\" is known as a **trial**.\n",
    "\n",
    "A **random variable** is a function that maps each outcome to a number. For example, the number of successes in 3 trials is a random variable, with the following mapping:\n",
    "\n",
    "```\n",
    "000 -> 0\n",
    "001 -> 1\n",
    "010 -> 1\n",
    "011 -> 2\n",
    "100 -> 1\n",
    "101 -> 2\n",
    "110 -> 2\n",
    "111 -> 3\n",
    "```\n",
    "\n",
    "**Count models** are concerned with the behavior of the ranodm variable tht can be interpreted as the number of successes in $k$ trials. In this notebook, we will explore three common count models - the binomial, Poisson and negative binomial distributions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Binomial model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Suppose we condcut 10 coin tossing trials. How many times do we see 0, 1, 2 ... successes? Intuitively, this number of successes $k$ can be no smaller than 0 and no larger than 10. Aslo, the number of scucesses will depend on whether the coin is fair or biased. Finally, for a fair coin, $k = 5$ is more likely to occur than $k = 0$ or $k = 10$. Why?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Fair coin"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "outcomes <- replicate(100, sum(sample(0:1, 10, replace=TRUE)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "options(repr.plot.width=4, repr.plot.height=3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {},
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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AIQiJAADjhCuIiGAAQgAAEIjEcABzwe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\nTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ggplot(data.frame(k=outcomes), aes(x=k)) +\n",
    "geom_bar() +\n",
    "scale_x_continuous(limits=c(0, 10), breaks=0:10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Biased coin"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "p <- 0.7\n",
    "outcomes <- replicate(100, sum(sample(0:1, 10, replace=TRUE, prob = c(1-p, p))))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {},
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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eeee8698cYbnUzXwvsPPPCA/36vFnAPC9Z3upADEIAABCAAgX2cQEEOWF393PWY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\nIACBshL4f56i1bsP7Rc5AAAAAElFTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ggplot(data.frame(k=outcomes), aes(x=k)) +\n",
    "geom_bar() +\n",
    "scale_x_continuous(limits=c(0, 10), breaks=0:10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The binomial distribution\n",
    "\n",
    "The two simulations above are examples of draws from the binomial distribtion. Here we repeat the simulaitons using the more compact binomial distribution model."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Fair coin"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "outcomes <- rbinom(n=100, size=10, prob=0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {},
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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at2+CgdOnQw/fNRevTo4aMaV0fDfotS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\ntAtDEUAAAQQQyKnA/wMf4f9VXH70GAAAAABJRU5ErkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ggplot(data.frame(k=outcomes), aes(x=k)) +\n",
    "geom_bar() +\n",
    "scale_x_continuous(limits=c(0, 10), breaks=0:10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Biased coin"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "outcomes <- rbinom(n=100, size=10, prob=0.7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {},
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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QhAoHQI4IBLJ6/QFAIQgAAEyogADriM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\ngg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ggplot(data.frame(k=outcomes), aes(x=k)) +\n",
    "geom_bar() +\n",
    "scale_x_continuous(limits=c(0, 10), breaks=0:10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exact theoretical distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "k <- 0:10\n",
    "n <- dbinom(x = k, size =10, prob = 0.7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {},
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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cZTLcB79+41uqbbeYL1fv369UWXldbW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\nrkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ggplot(data.frame(k=k, n=n), aes(x=k, y=n)) +\n",
    "geom_bar(stat=\"identity\") +\n",
    "scale_x_continuous(limits=c(0, 10), breaks=0:10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 1**\n",
    "\n",
    "Make a histogram of the number of successes expected from 5 trials when the probability of success in each trial is 0.25."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 2**\n",
    "\n",
    "You have a miracle drug that will cure 70% of people infected with HIV. You run many clinical trials with 100 HIV-infected people in each trial using this drug. What fraction of the trials have 80 or more people cured?\n",
    "\n",
    "- Find the answer using properties of the binomial distribution\n",
    "- Find the answer using a simulation with 10000 trials"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Mean and variance of the binomial distribution\n",
    "\n",
    "The mean of the binomial distribuiton is $np$ and the variance is $np(1-p)$. Note that the mean and variance are defined by the same two parameters $n$ and $p$ - once the mean is known, the variance is fixed. Hence it is common for real world data to resemble a binomial distribuiotn except that the variance is larger than expected - this is known as over-dispersion."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "n <- 10\n",
    "p <- 0.7\n",
    "x <- rbinom(10000, n, p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "6.9922"
      ],
      "text/latex": [
       "6.9922"
      ],
      "text/markdown": [
       "6.9922"
      ],
      "text/plain": [
       "[1] 6.9922"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mean(x)"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "n * p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "2.08"
      ],
      "text/latex": [
       "2.08"
      ],
      "text/markdown": [
       "2.08"
      ],
      "text/plain": [
       "[1] 2.08"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "round(var(x), 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "2.1"
      ],
      "text/latex": [
       "2.1"
      ],
      "text/markdown": [
       "2.1"
      ],
      "text/plain": [
       "[1] 2.1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n * p * (1-p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Poisson distribution\n",
    "\n",
    "When the number of trials $n$ is very large, and the probability of success $p$ is very small, the binomial distribution can be aproxiated by the simpler Poisson distribution which only has a single parameter $\\lambda = np$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>x</th><th scope=col>binomial</th><th scope=col>poisson</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>0     </td><td>0.3677</td><td>0.3679</td></tr>\n",
       "\t<tr><td>1     </td><td>0.3681</td><td>0.3679</td></tr>\n",
       "\t<tr><td>2     </td><td>0.1840</td><td>0.1839</td></tr>\n",
       "\t<tr><td>3     </td><td>0.0613</td><td>0.0613</td></tr>\n",
       "\t<tr><td>4     </td><td>0.0153</td><td>0.0153</td></tr>\n",
       "\t<tr><td>5     </td><td>0.0030</td><td>0.0031</td></tr>\n",
       "\t<tr><td>6     </td><td>0.0005</td><td>0.0005</td></tr>\n",
       "\t<tr><td>7     </td><td>0.0001</td><td>0.0001</td></tr>\n",
       "\t<tr><td>8     </td><td>0.0000</td><td>0.0000</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|lll}\n",
       " x & binomial & poisson\\\\\n",
       "\\hline\n",
       "\t 0      & 0.3677 & 0.3679\\\\\n",
       "\t 1      & 0.3681 & 0.3679\\\\\n",
       "\t 2      & 0.1840 & 0.1839\\\\\n",
       "\t 3      & 0.0613 & 0.0613\\\\\n",
       "\t 4      & 0.0153 & 0.0153\\\\\n",
       "\t 5      & 0.0030 & 0.0031\\\\\n",
       "\t 6      & 0.0005 & 0.0005\\\\\n",
       "\t 7      & 0.0001 & 0.0001\\\\\n",
       "\t 8      & 0.0000 & 0.0000\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "x | binomial | poisson | \n",
       "|---|---|---|---|---|---|---|---|---|\n",
       "| 0      | 0.3677 | 0.3679 | \n",
       "| 1      | 0.3681 | 0.3679 | \n",
       "| 2      | 0.1840 | 0.1839 | \n",
       "| 3      | 0.0613 | 0.0613 | \n",
       "| 4      | 0.0153 | 0.0153 | \n",
       "| 5      | 0.0030 | 0.0031 | \n",
       "| 6      | 0.0005 | 0.0005 | \n",
       "| 7      | 0.0001 | 0.0001 | \n",
       "| 8      | 0.0000 | 0.0000 | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "  x binomial poisson\n",
       "1 0 0.3677   0.3679 \n",
       "2 1 0.3681   0.3679 \n",
       "3 2 0.1840   0.1839 \n",
       "4 3 0.0613   0.0613 \n",
       "5 4 0.0153   0.0153 \n",
       "6 5 0.0030   0.0031 \n",
       "7 6 0.0005   0.0005 \n",
       "8 7 0.0001   0.0001 \n",
       "9 8 0.0000   0.0000 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x <- 0:8\n",
    "n <- 1000\n",
    "p <- 0.001\n",
    "binomial <- dbinom(x, size=n, prob=p)\n",
    "poisson <-  dpois(x, lambda=n*p)\n",
    "round(data.frame(x=x, binomial=binomial, poisson=poisson), 4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 3**\n",
    "\n",
    "Suppose that RNA-seq results in calling errors (e.g. an A is read as a C) once every 1,000 base pairs, and generated single-end reads of exactly 200 base pairs. What fraction of reads would have more than 2 errors?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Mean and varinace of the Poisson distribuiotn\n",
    "\n",
    "The mean of the Poisson distribution is $\\lambda$, and its variance is also $\\lambda$. Hence, just as for the binomila distribution, it is common to find real-world data that resembles the Posson distribution, except that the variance is larger than expected. This is another example of over-dispersion."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x <- rpois(10000, 3.14)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "3.15"
      ],
      "text/latex": [
       "3.15"
      ],
      "text/markdown": [
       "3.15"
      ],
      "text/plain": [
       "[1] 3.15"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "round(mean(x), 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "3.17"
      ],
      "text/latex": [
       "3.17"
      ],
      "text/markdown": [
       "3.17"
      ],
      "text/plain": [
       "[1] 3.17"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "round(var(x), 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## The negative binomial distribution\n",
    "\n",
    "There are two ways $k$ successes from a series of $n$ Bernoulli trials can arise. In the first way, there were $n$ trials planned, and it just so happened that $k$ of these were successes. This is modeled by the binomial distribution. However, an alternative scenario is that trials are run until exactly $k$ successes are observed. This is modeled by the negative binomial distribution. \n",
    "\n",
    "Note that the `nbinom` family of functions in R models the number of **failures** before a target number of successes is reached, and not the number of trials."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "binom <- rbinom(n=100, size=10, prob=0.5)\n",
    "nbinom <- rnbinom(n=100, size=10, prob=0.5)\n",
    "df <- data.frame(binom=binom, nbinom=nbinom)\n",
    "df <- df %>% gather(dist, count)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {},
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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74wVSRAAiRAAh4nQAPs8QymeiRAAiRA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\n6ZdnTDEJkAAJkIAHCPx/dZCoC+3o3fEAAAAASUVORK5CYII=",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ggplot(df, aes(x=count, color=dist, fill=dist)) + \n",
    "facet_wrap(~ dist) +\n",
    "geom_bar(alpha=0.5) +\n",
    "guides(color=F, fill=F)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Utility functions for alternative parameteizations of negative binomial"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "nb.mean <- function(r, p) {\n",
    "    r * (1-p)/p\n",
    "}\n",
    "\n",
    "nb.var <- function(r, p) {\n",
    "    r * (1-p)/p^2\n",
    "}\n",
    "\n",
    "nb.r <- function(mu, s2) {\n",
    "    mu^2/(s2 - mu)\n",
    "}\n",
    "\n",
    "nb.p <- function(mu, s2) {\n",
    "    mu/s2\n",
    "}\n",
    "\n",
    "nb.disp <- function(mu, s2) {\n",
    "    (1/mu)*(s2/mu - 1)\n",
    "}\n",
    "\n",
    "nb.disp2 <- function(r, p) {\n",
    "    mu <- nb.mean(r, p)\n",
    "    s2 <- nb.var(r, p)\n",
    "    nb.disp(mu, s2)\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
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      "text/plain": [
       "[1] 10"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "nb.mean(10, 0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "20"
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       "[1] 20"
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     },
     "metadata": {},
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    }
   ],
   "source": [
    "nb.var(10, 0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.1"
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       "0.1"
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       "0.1"
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       "[1] 0.1"
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     "metadata": {},
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   ],
   "source": [
    "nb.disp2(10, 0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x <- rnbinom(n=100000, size=10, prob=0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "10"
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      "text/latex": [
       "10"
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      "text/markdown": [
       "10"
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      "text/plain": [
       "[1] 10"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "round(mean(x), 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "20"
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       "20"
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       "20"
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      "text/plain": [
       "[1] 20"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "round(var(x), 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 4**\n",
    "\n",
    "Set $\\lambda = 1$ for the Poisson distribution, and $\\mu = $ for the negative binomial distribution. Compare the two distributions when $\\alpha = 0.001$ and $\\alpha = 1.0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The multinomial distribution\n",
    "\n",
    "The binomial distribtion only allows two outcomes traditionally known as success and failure. The multinomial distiribution generalizes the binomial to allow more than two outcomes. \n",
    "\n",
    "A common model for the multinomial distribution is that of distributing $n$ balls to $k$ urns."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>2</td><td>1</td><td>2</td><td>2</td><td>1</td><td>1</td><td>0</td><td>2</td><td>2</td><td>0</td></tr>\n",
       "\t<tr><td>1</td><td>2</td><td>2</td><td>2</td><td>2</td><td>3</td><td>2</td><td>1</td><td>2</td><td>2</td></tr>\n",
       "\t<tr><td>4</td><td>1</td><td>1</td><td>2</td><td>3</td><td>2</td><td>5</td><td>5</td><td>3</td><td>1</td></tr>\n",
       "\t<tr><td>3</td><td>6</td><td>5</td><td>4</td><td>4</td><td>4</td><td>3</td><td>2</td><td>3</td><td>7</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{llllllllll}\n",
       "\t 2 & 1 & 2 & 2 & 1 & 1 & 0 & 2 & 2 & 0\\\\\n",
       "\t 1 & 2 & 2 & 2 & 2 & 3 & 2 & 1 & 2 & 2\\\\\n",
       "\t 4 & 1 & 1 & 2 & 3 & 2 & 5 & 5 & 3 & 1\\\\\n",
       "\t 3 & 6 & 5 & 4 & 4 & 4 & 3 & 2 & 3 & 7\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| 2 | 1 | 2 | 2 | 1 | 1 | 0 | 2 | 2 | 0 | \n",
       "| 1 | 2 | 2 | 2 | 2 | 3 | 2 | 1 | 2 | 2 | \n",
       "| 4 | 1 | 1 | 2 | 3 | 2 | 5 | 5 | 3 | 1 | \n",
       "| 3 | 6 | 5 | 4 | 4 | 4 | 3 | 2 | 3 | 7 | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "     [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]\n",
       "[1,] 2    1    2    2    1    1    0    2    2    0    \n",
       "[2,] 1    2    2    2    2    3    2    1    2    2    \n",
       "[3,] 4    1    1    2    3    2    5    5    3    1    \n",
       "[4,] 3    6    5    4    4    4    3    2    3    7    "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rmultinom(10, size=10, prob=c(1,2,3,4)/10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "**Exercise 5** \n",
    "\n",
    "Suppose the genome has 20,000 genes and we sequence 1 million reads. If each read is equally likely to be mapped to any gene\n",
    "\n",
    "- What is the average number of reads mapped to each gene?\n",
    "- What is the standard deviation of the number of reads mapped to each gene?\n",
    "- what is the 95% confidence interval for this estimate of the mean?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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