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   "source": [
    "# Lecture 9: Expectation, Indicator Random Variables, Linearity\n",
    "\n",
    "\n",
    "## Stat 110, Prof. Joe Blitzstein, Harvard University\n",
    "\n",
    "----"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## More on Cumulative Distribution Functions\n",
    "\n",
    "A CDF: $F(x) = P(X \\le x)$, as a function of real $x$ has to be\n",
    "\n",
    "* non-negative\n",
    "* add up to 1\n",
    "\n",
    "In the following discrete case, it is easy to see how the probability mass function (PMF) relates to the CDF:\n",
    "\n",
    "![title](images/L0901.png)\n",
    "\n",
    "Therefore, you can compute any probability given a CDF. \n",
    "\n",
    "_Ex. Find $P(1 \\lt x \\le 3)$ using CDF $F$._\n",
    "\n",
    "\\begin{align}\n",
    "  & &P(x \\le 1) + P(1 \\lt x \\le 3) &= P(x \\le 3) \\\\\n",
    "  & &\\Rightarrow P(1 \\lt x \\le 3) &= F(3) - F(1)\n",
    "\\end{align}\n",
    "\n",
    "Note that while we don't need to be so strict in the __continuous case__, for the discrete case you need to be careful about the $\\lt$ and $\\le$.\n",
    "\n",
    "\n",
    "### Properties of CDF\n",
    "\n",
    "A function $F$ is a CDF __iff__ the following three conditions are satisfied.\n",
    "1. increasing\n",
    "1. right-continuous (function is continuous as you _approach a point from the right_)\n",
    "1. $F(x) \\rightarrow 0 \\text{ as } x \\rightarrow - \\infty$, and $F(x) \\rightarrow 1 \\text{ as } x \\rightarrow \\infty$.\n",
    "\n",
    "----"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Independence of Random Variables\n",
    "\n",
    "$X, Y$ are independent r.v. if\n",
    "\n",
    "\\begin{align}\n",
    "  \\underbrace{P(X \\le x, Y \\le y)}_{\\text{joint CDF}} &= P(X \\le x) P(Y \\le y) & &\\text{ for all x, y in the continuous case} \\\\\n",
    "  \\\\\n",
    "  \\underbrace{P(X=x, Y=y)}_{\\text{joint PMF}} &= P(X=x) P(Y=y) & &\\text{ for all x, y in the discrete case}\n",
    "\\end{align}\n",
    "\n",
    "----"
   ]
  },
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   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Averages of Random Variables (mean, Expected Value)\n",
    "\n",
    "A mean is... well, the _average of a sequence of values_.\n",
    "\n",
    "\\begin{align}\n",
    "  1, 2, 3, 4, 5, 6 \\rightarrow \\frac{1+2+3+4+5+6}{6} = 3.5\n",
    "\\end{align}\n",
    "\n",
    "In the case where there is repetition in the sequence\n",
    "\n",
    "\\begin{align}\n",
    "  1,1,1,1,1,3,3,5 \\rightarrow & \\frac{1+1+1+1+1+3+3+5}{8} \\\\\n",
    "  \\\\\n",
    "  & \\dots \\text{ or } \\dots \\\\\n",
    "  \\\\\n",
    "  & \\frac{5}{8} ~~ 1 + \\frac{2}{8} ~~ 3 + \\frac{1}{8} ~~ 5 & &\\quad \\text{ ... weighted average}\n",
    "\\end{align}\n",
    "\n",
    "where the weights are the frequency (fraction) of the unique elements in the sequence, and these weights add up to 1.\n",
    "\n",
    "### Expected value of a discrete r.v. $X$\n",
    "\n",
    "\\begin{align}\n",
    "  \\mathbb{E}(X) = \\sum_{x} \\underbrace{x}_{\\text{value}} ~~ \\underbrace{P(X=x)}_{\\text{PMF}} ~& &\\quad \\text{ ... summed over x with  } P(X=x) \\gt 0\n",
    "\\end{align}\n",
    "\n",
    "### Expected value of  $X \\sim \\operatorname{Bern}(p)$\n",
    "\n",
    "\\begin{align}\n",
    "  \\text{Let } X &\\sim \\operatorname{Bern}(p) \\\\\n",
    "  \\mathbb{E}(X) &= \\sum_{k=0}^{1} k P(X=k) \\\\\n",
    "  &= 1 ~~ P(X=1) + 0 ~~ P(X=0) \\\\\n",
    "  &= p\n",
    "\\end{align}\n",
    "\n",
    "### Expected value of an Indicator Variable\n",
    "\n",
    "\\begin{align}\n",
    "  X &=\n",
    "  \\begin{cases}\n",
    "    1, &\\text{ if A occurs} \\\\\n",
    "    0, &\\text{ otherwise }\n",
    "  \\end{cases} \\\\\n",
    "  \\\\\n",
    "  \\therefore \\mathbb{E}(X) &= P(A)\n",
    "\\end{align}\n",
    "\n",
    "Notice how this lets us relate (bridge) the expected value $\\mathbb{E}(X)$ with a probability $P(A)$.\n",
    "\n",
    "#### Average of $X \\sim \\operatorname{Bin}(n,p)$\n",
    "\n",
    "There is a hard way to do this, and an easy way.\n",
    "\n",
    "First the hard way:\n",
    "\n",
    "\\begin{align}\n",
    "  \\mathbb{E}(X) &= \\sum_{k=0}^{n} k \\binom{n}{k} p^k (1-p)^{n-k} \\\\\n",
    "    &= \\sum_{k=0}^{n} n \\binom{n-1}{k-1} p^k (1-p)^{n-k} & &\\text{from Lecture 2, Story proofs, ex. 2, choosing a team and president} \\\\\n",
    "    &= np \\sum_{k=0}^{n} n \\binom{n-1}{k-1} p^{k-1} (1-p)^{n-k} \\\\\n",
    "    &= np \\sum_{j=0}^{n-1} \\binom{n-1}{j} p^j(1-p)^{n-1-j} & &\\text{letting } j=k-1 \\text{, which sets us up to use the Binomial Theorem} \\\\\n",
    "    &= np\n",
    "\\end{align}\n",
    "\n",
    "Now, what about the _easy way_?\n",
    "\n",
    "----"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Linearity of Expected Values\n",
    "\n",
    "Linearity is this:\n",
    "\n",
    "\\begin{align}\n",
    "  \\mathbb{E}(X+Y) &= \\mathbb{E}(X) + \\mathbb{E}(Y) & &\\quad \\text{even if X and Y are dependent}\\\\\n",
    "  \\\\\n",
    "  \\mathbb{E}(cX) &= c \\mathbb{E}(X)\\\\\n",
    "\\end{align}\n",
    "\n",
    "\n",
    "### Expected value of Binomial r.v using Linearity\n",
    "\n",
    "Let $X \\sim \\operatorname{Bin}(n,p)$. The easy way to calculate the expected value of a binomial r.v. follows.\n",
    "\n",
    "Let $X = X_1 + X_2 + \\dots + X_n$ where $X_j \\sim \\operatorname{Bern}(P)$.\n",
    "\n",
    "\\begin{align}\n",
    "  \\mathbb{E}(X) &= \\mathbb{E}(X_1 + X_2 + \\dots + X_n) \\\\\n",
    "  \\mathbb{E}(X) &= \\mathbb{E}(X_1) + \\mathbb{E}(X_2) + \\dots + \\mathbb{E}(X_n) & &\\quad \\text{by Linearity}\\\\\n",
    "  \\mathbb{E}(X) &= n \\mathbb{E}(X_1) & &\\quad \\text{by symmetry}\\\\\n",
    "  \\mathbb{E}(X) &= np \n",
    "\\end{align}\n",
    "\n",
    "\n",
    "### Expected value of  Hypergeometric r.v.\n",
    "\n",
    "Ex. 5-card hand $X=(\\# aces)$. Let $X_j$ be the indicator that the $j^{th}$ card is an ace.\n",
    "\n",
    "\\begin{align}\n",
    "  \\mathbb{E}(X) &= \\mathbb{E}(X_1 + X_2 + X_3 + X_4 + X_5) \\\\\n",
    "  &= \\mathbb{E}(X_1) + \\mathbb{E}(X_2) + \\mathbb{E}(X_3) + \\mathbb{E}(X_4) + \\mathbb{E}(X_5) & &\\quad \\text{by Linearity} \\\\\n",
    "  &= 5 ~~ \\mathbb{E}(X_1) & &\\quad \\text{by symmetry} \\\\\n",
    "  &= 5 ~~ P(1^{st} \\text{ card is ace}) & &\\quad \\text{by the Fundamental Bridge}\\\\\n",
    "  &= \\boxed{\\frac{5}{13}}\n",
    "\\end{align}\n",
    "\n",
    "Note that when we use linearity in this case, the individual probabilities are _weakly dependent_, in that the probability of getting an ace decreases slightly; and that if you already have four aces, then the fifth card cannot possibly be an ace. But using linearity, we can nevertheless quickly and easily compute $\\mathbb{E}(X_1 + X_2 + X_3 + X_4 + X_5)$.\n",
    "\n",
    "----"
   ]
  },
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   },
   "source": [
    "## Geometric Distribution\n",
    "\n",
    "### Description\n",
    "\n",
    "The Geometric distribution comprises a series of independent $\\operatorname{Bern}(p)$ trials where we count the number of failures before the first success. \n",
    "\n",
    "### Notation\n",
    "\n",
    "$X \\sim \\operatorname{Geom}(p)$.\n",
    "\n",
    "### Parameters\n",
    "\n",
    "$0 < p < 1 \\text{, } p \\in \\mathbb{R}$"
   ]
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  {
   "cell_type": "code",
   "execution_count": 1,
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MEXrkOZVKBQB+2/N11113Cf9v3rw53n33XeFvs9mM2NhY4e/NmzcjMTERgwcPrmq2Cr3x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LJibZLMWJ8kq3pyZ1MFdNu2YZr8xkySF69EQLJibZLMWJ8ko3pyZ9MEdN/3kc1mYVlW1E0hIiIi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\nUZGUUvTP74etbY8fP+72/ejpmFwuR3t7O06ePImioiLExcVBL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      "text/plain": [
       "<matplotlib.figure.Figure at 0x295f500a1d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "from matplotlib.ticker import (MultipleLocator, FormatStrFormatter,\n",
    "                               AutoMinorLocator)\n",
    "from scipy.stats import geom\n",
    "\n",
    "%matplotlib inline\n",
    "\n",
    "plt.xkcd()\n",
    "_, ax = plt.subplots(figsize=(12,8))\n",
    "\n",
    "# seme Geometric parameters\n",
    "p_values = [0.2, 0.5, 0.75]\n",
    "\n",
    "# colorblind-safe, qualitative color scheme\n",
    "colors = ['#1b9e77', '#d95f02', '#7570b3']\n",
    "\n",
    "for i,p in enumerate(p_values):\n",
    "    x = np.arange(geom.ppf(0.01, p), geom.ppf(0.99, p))\n",
    "    pmf = geom.pmf(x, p)\n",
    "    ax.plot(x, pmf, 'o', color=colors[i], ms=8, label='p={}'.format(p))\n",
    "    ax.vlines(x, 0, pmf, lw=2, color=colors[i], alpha=0.3)\n",
    "\n",
    "# legend styling\n",
    "legend = ax.legend()\n",
    "for label in legend.get_texts():\n",
    "    label.set_fontsize('large')\n",
    "for label in legend.get_lines():\n",
    "    label.set_linewidth(1.5)\n",
    "\n",
    "# y-axis\n",
    "ax.set_ylim([0.0, 0.9])\n",
    "ax.set_ylabel(r'$P(X=k)$')\n",
    "\n",
    "# x-axis\n",
    "ax.set_xlim([0, 20])\n",
    "ax.set_xlabel('# of failures k before first success')\n",
    "\n",
    "# x-axis tick formatting\n",
    "majorLocator = MultipleLocator(5)\n",
    "majorFormatter = FormatStrFormatter('%d')\n",
    "minorLocator = MultipleLocator(1)\n",
    "ax.xaxis.set_major_locator(majorLocator)\n",
    "ax.xaxis.set_major_formatter(majorFormatter)\n",
    "ax.xaxis.set_minor_locator(minorLocator)\n",
    "\n",
    "ax.grid(color='grey', linestyle='-', linewidth=0.3)\n",
    "\n",
    "plt.suptitle(r'Geometric PMF: $P(X=k) = pq^k$')\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Probability mass function\n",
    "\n",
    "Consider the event $A$ where there are 5 failures before the first success. We could notate this event $A$ as $\\text{FFFFFS}$, where $F$ denotes failure and $S$ denotes the first success. Note that this string **must** end with a success. So, $P(A) = q^5p$.\n",
    "\n",
    "And from just this, we can derive the PMF for a geometric r.v.\n",
    "\n",
    "\\begin{align}\n",
    "  P(X=k) &= pq^k \\text{, } k \\in \\{1,2, \\dots \\}\n",
    "  \\\\\n",
    "  \\\\\n",
    "  \\sum_{k=0}^{\\infty} p q^k &= p \\sum_{k=0}^{\\infty} q^k \\\\\n",
    "  &= p ~~ \\frac{1}{1-q} & &\\quad \\text{by the geometric series where } |r| < 1 \\\\\n",
    "  &= \\frac{p}{p} \\\\\n",
    "  &= 1 & &\\quad \\therefore \\text{ this is a valid PMF}\n",
    "\\end{align} \n",
    "\n",
    "### Expected value\n",
    "\n",
    "So, the hard way to calculate the expected value $\\mathbb{E}(X)$ of a $\\operatorname{Geom}(p)$ is\n",
    "\n",
    "\\begin{align}\n",
    "  \\mathbb{E}(X) &= \\sum_{k=0}^{\\infty} k p q^k \\\\\n",
    "  &= p \\sum_{k=0}^{\\infty} k q^k \\\\\n",
    "  \\\\\n",
    "  \\\\\n",
    "  \\text{ now ... } \\sum_{k=0}^{\\infty} q^k &=  \\frac{1}{1-q} & &\\quad \\text{by the geometric series where |q| < 1} \\\\\n",
    "  \\sum_{k=0}^{\\infty} k q^{k-1} &= \\frac{1}{(1-q)^2} & &\\quad \\text{by differentiating with respect to k} \\\\\n",
    "  \\sum_{k=0}^{\\infty} k q^{k} &= \\frac{q}{(1-q)^2} \\\\\n",
    "  &= \\frac{q}{p^2} \\\\\n",
    "  \\\\\n",
    "  \\\\\n",
    "  \\text{ and returning, we have ... } \\mathbb{E}(X) &= p ~~ \\frac{q}{(p^2} \\\\\n",
    "  &= \\frac{q}{p} & &\\quad \\blacksquare\n",
    "\\end{align}\n",
    "\n",
    "And here is the story proof, without using the geometric series and derivatives:\n",
    "\n",
    "Again, we are considering a series of independent Bernoulli trials with probability of success $p$, and we are counting the number of failures before getting the first success.\n",
    "\n",
    "Similar to doing a first step analysis in the case of the Gambler's Ruin, we look at the first case where we either:\n",
    "\n",
    "* get a heads (success) on the very first try, meaning 0 failures\n",
    "* or we get 1 failure, but we start the process all over again\n",
    "\n",
    "Remember that in the case of a coin flip, the coin has no memory.\n",
    "\n",
    "Let $c=\\mathbb{E}(X)$.\n",
    "\n",
    "\\begin{align}\n",
    "  c &= 0 ~~ p + (1 + c) ~~ q \\\\\n",
    "    &= q + qc \\\\\n",
    "  \\\\\n",
    "  c - cq &= q \\\\\n",
    "  c (1 - q) &= q \\\\\n",
    "  c &= \\frac{q}{1-q} \\\\\n",
    "    &= \\frac{q}{p} & &\\quad \\blacksquare\n",
    "\\end{align}\n",
    "\n",
    "----"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "View [Lecture 9: Expectation, Indicator Random Variables, Linearity | Statistics 110](http://bit.ly/2nPbrNe) on YouTube."
   ]
  }
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