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    "# Lecture 19: Joint, Conditional and Marginal Distributions; 2D LOTUS; Expected Distance between Uniforms; Chicken-egg Problem\n",
    "\n",
    "\n",
    "## Stat 110, Prof. Joe Blitzstein, Harvard University\n",
    "\n",
    "----"
   ]
  },
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   "metadata": {},
   "source": [
    "### Joint, Conditional and Marginal Distributions\n",
    "\n",
    "#### Joint CDF\n",
    "A joint CDF is simply where we are dealing with multiple random variables. As an example, a case where we have two random variables $X, Y$, the joint CDF of two random variables $X, Y$ can be expressed as:\n",
    "\n",
    "\\begin{align}\n",
    "  F(x,y) &= P(X {\\le} x, Y {\\le} y)\n",
    "\\end{align}\n",
    "\n",
    "Note that the random variables may be discrete, continuous, or a mixture of both.\n",
    "\n",
    "#### Joint PDF\n",
    "The joint PDF, in the case of _continuous_ random variables, is what you would _integrate_ to get the joint CDF. Continuing with our example of two (continous) random variables, we have:\n",
    "\n",
    "\\begin{align}\n",
    "  f(x,y) &= \\frac{\\partial^2}{\\partial{x}\\partial{y}} F(x,y)\n",
    "\\end{align}\n",
    "\n",
    "Conversely, if we want to know the probability of $X,Y$ in some set $A$, we _integrate_ the density to get that probability.\n",
    "\n",
    "\\begin{align}\n",
    "  P\\left((X,Y) \\in A\\right) &= \\iint\\limits_{A} f(x,y) \\, dxdy\n",
    "\\end{align}\n",
    "\n",
    "Integrate by holding one variable constant, and then do the other. The key thing is to be sure to get the _limits of integration_ correct.\n",
    "\n",
    "#### Marginal PDF\n",
    "The _marginal PDF of $X$_ is obtained by _integrating out the $Y$_. Recall the $X,Y$ contigency table and the definition of marginal probability.\n",
    "\n",
    "\\begin{align}\n",
    "  \\int_{-\\infty}^{\\infty} f(x,y) \\, dy\n",
    "\\end{align}\n",
    "\n",
    "Notice that by keeping $X$ constant and _integrating over all $Y$_, the marginal PDF of $X$ no longer depends on $Y$.\n",
    "\n",
    "And we can do vice-versa for the marginal PDF of $Y$, but keeping $Y$ constant and _integrating over all $X$_.\n",
    "\n",
    "Do not forget that taking the marginal PDF of $X$ and then integrating over all $X$, we should get 1.0.\n",
    "\n",
    "\\begin{align}\n",
    "  \\int\\limits_{-\\infty}^{\\infty} \\int\\limits_{-\\infty}^{\\infty} f(x,y) \\, dx dy &= 1.0\n",
    "\\end{align}\n",
    "\n",
    "#### Conditional PDF\n",
    "\n",
    "Given that we know $X$, what is the appropriate PDF for $Y$?\n",
    "\n",
    "Well, we can apply what we know about _conditional probability_ to get a conditional PDF.\n",
    "\n",
    "\\begin{align}\n",
    "  f_{Y|X} (y|x) &= \\frac{f_{XY}(x,y)}{f_{X}(x)} \\\\\n",
    "    &= \\frac{f_{X|Y}(x|y) \\, f_{Y}(y)}{f_{X}(x)}\n",
    "\\end{align}\n",
    "\n",
    "This is completely analogous to conditional probability. \n",
    "\n",
    "#### Independence\n",
    "\n",
    "$X,Y$ are independent if\n",
    "\n",
    "\\begin{align}\n",
    "  f_{X,Y}(x,y) &= f_{X}(x) \\, f_{Y}(y) &\\quad \\text{for all }x, y \\text{ from PDF p.o.v.} \\\\\n",
    "  \\\\\n",
    "  F(x,y) &= F(x) \\, F(y) &\\quad \\text{for all }x, y \\text{ from CDF p.o.v.} \n",
    "\\end{align}\n",
    "\n",
    "These statements are equivalent, but in most cases it might be easier to work the PDFs.\n",
    "\n",
    "----"
   ]
  },
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   "cell_type": "markdown",
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   },
   "source": [
    "### A Uniform example\n",
    "\n",
    "Let's revisit that distribution that is uniform on the unit disc $x^2 + y^2 \\le 1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
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sVLzSlgUJ6uOPP46hoaFZJQarV69GvV7H6OjorL8ZHh7Grl27UCqVcOONN+KK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\ngioiIiIyRvwPb6jzBkGSwLUAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1d107651c18>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline  \n",
    "\n",
    "plt.xkcd()\n",
    "\n",
    "fig = plt.figure(figsize = (5.0,5.0))\n",
    "ax = fig.add_subplot(1, 1, 1)\n",
    "ax.set_xlim([-1.5,1.5])\n",
    "ax.set_ylim([-1.5,1.5])\n",
    "circ = plt.Circle((0, 0), radius=1.0, color=\"b\", alpha=0.2, lw=5)\n",
    "ax.add_patch(circ)\n",
    "plt.axhline(0, color=\"black\", alpha=0.4)\n",
    "plt.axvline(0, color=\"black\", alpha=0.4)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Joint PDF\n",
    "\n",
    "A valid PDF is required to integrate to 1.0, and since we are looking at a Uniform distribution over the unit circle with area equal to $\\pi$,  we have\n",
    "\n",
    "\\begin{align}\n",
    "  f_{XY}(x,y) &= \n",
    "    \\begin{cases}\n",
    "      \\frac{1}{\\pi}  & \\quad \\text{if } x^2 + y^2 \\le 1 \\\\\n",
    "      0  & \\quad \\text{otherwise}\\\\\n",
    "    \\end{cases}\n",
    "\\end{align}\n",
    "\n",
    "Understand that while this is uniformly distributed, $x$ and $y$ are closely related with $x^2 + y^2 \\le 1$, and so $X,Y$ cannot be independent. More on that coming up.\n",
    "\n",
    "#### Marginal PDF of $X$\n",
    "\n",
    "\\begin{align}\n",
    "  f_{X}(x) &= \\int_{-\\sqrt{1-x^2}}^{\\sqrt{1-x^2}} \\frac{1}{\\pi} dy \\\\\n",
    "  \\\\\n",
    "  &= \\frac{2}{\\pi} \\sqrt{1-x^2} &\\quad \\text{where } -1 \\le x \\le 1\n",
    "\\end{align}\n",
    "\n",
    "Notice that while the joint PDF is Uniform, the marginal PDF is not Uniform: it grows as $x$ approaches 0. \n",
    "\n",
    "By _symmetry_ we can just replace $x$ with $y$ to get the marginal PDF of $Y$ $f_{Y}(y)$\n",
    "\n",
    "#### Conditional PDF of $Y|X$\n",
    "\n",
    "\\begin{align}\n",
    "  f_{Y|X}(y|x) &= \\frac{f_{XY}(x,y)}{f_{X}(x)} \\\\\n",
    "  &= \\frac{\\frac{1}{\\pi}}{\\frac{2}{\\pi} \\sqrt{1-x^2}} \\\\\n",
    "  &= \\frac{1}{2 \\sqrt{1-x^2}} &\\quad \\text{if } -\\sqrt{1-x^2} \\le y \\le \\sqrt{1-x^2}\n",
    "\\end{align}\n",
    "\n",
    "But since we are treating $x$ as a constant, and the above equation does not _depend on $y$_, the conditional PDF $f_{Y|X}(y|x)$ is actually Uniform.\n",
    "\n",
    "\\begin{align}\n",
    "  Y|X &\\sim \\operatorname{Unif}(-\\sqrt{1-x^2}, \\sqrt{1-x^2}) &\\quad \\text{or also } \\\\\n",
    "  \\\\\n",
    "  Y|X=x \\, &\\sim \\operatorname{Unif}(-\\sqrt{1-x^2}, \\sqrt{1-x^2})\n",
    "\\end{align}\n",
    "\n",
    "#### Non-independence\n",
    "\n",
    "Since in our example above\n",
    "\n",
    "\\begin{align}\n",
    "  f_{XY}(x,y) \\neq f_{X}(x) \\, f_{Y}(y)\n",
    "\\end{align}\n",
    "\n",
    "we can see that $X,Y$ are not independent.\n",
    "\n",
    "Or, because the _unconditional_ distribution of $Y$ is not the same as the _conditional_ distribution of $Y|X$, it necessarily follows that $X,Y$ are not independent as learning $X$ gives us information about $Y$ (and vice-versa).\n",
    "\n",
    "\n",
    "----"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2D LOTUS\n",
    "\n",
    "Let $X,Y$ have joint PDF $f(x,y)$, and let $g(x,y)$ be a real-valued function of $x,y$.\n",
    "\n",
    "Then, without trying to find the PDF of $g(x,y)$, LOTUS lets us use joint PDF:\n",
    "\n",
    "\\begin{align}\n",
    "  \\mathbb{E} g(x,y) &= \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} g(x,y) \\, f(x,y) \\, dx dy\n",
    "\\end{align}\n",
    "\n",
    "We will illustrate this by proving the following theorem.\n",
    "\n",
    "#### Theorem\n",
    "If $X,Y$ are independent, then $\\mathbb{E}(XY) = \\mathbb{E}(X)\\mathbb{E}(Y)$.\n",
    "\n",
    "Or, in other words, the _independence_ of r.v. $X$ and $Y$ implies that they are _uncorrelated_.\n",
    "\n",
    "#### Proof\n",
    "\n",
    "Consider the continuous case\n",
    "\n",
    "\\begin{align}\n",
    "  \\mathbb{E}(XY) &= \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} xy \\, f(x,y) \\, dx dy &\\quad \\text{2D LOTUS} \\\\\n",
    "  &= \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} xy \\, f_{X}(x) \\, f_{Y}(y) \\, dx dy &\\quad \\text{since } X,Y \\text{ are independent} \\\\\n",
    "  &= \\int_{-\\infty}^{\\infty} \\left( \\int_{-\\infty}^{\\infty} xy \\, f_{X}(x) \\, f_{Y}(y) \\, dx \\right) dy &\\quad \\text{this is a double integral, so we can group thusly} \\\\\n",
    "  &= \\int_{-\\infty}^{\\infty} y \\, f_{Y}(y) \\underbrace{ \\left( \\int_{-\\infty}^{\\infty} x \\, f_{X}(x) \\, dx \\right)}_{\\text{actually }\\mathbb{E}(X) \\text{, which is constant}} dy &\\quad \\text{since we are holding } y \\text{ constant} \\\\\n",
    "  &= \\mathbb{E}(X)  \\int_{-\\infty}^{\\infty} y \\, f_{Y}(y) \\, dy \\\\\n",
    "  &= \\mathbb{E}(X) \\, \\mathbb{E}(Y) &\\quad \\blacksquare\n",
    "\\end{align}\n",
    "\n",
    "----"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example: expected distance between 2 Uniformly distributed random points\n",
    "\n",
    "Given _i.i.d._ $X,Y \\sim Unif(0,1)$, find the expected distance between $X$ and $Y$, $\\mathbb{E}|X-Y|$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1d1080e0cc0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.xkcd()\n",
    "\n",
    "\n",
    "fig,ax = plt.subplots(1, figsize=(6.0,1.0))\n",
    "\n",
    "plt.plot([1/3, 2/3],[0,0], 'ro')\n",
    "ax.axhline(y=0, color='k', alpha=0.2)\n",
    "\n",
    "ax.set_xticks([0.0, (1.0/3.0), (2.0/3.0), 1.0])\n",
    "ax.set_xticklabels(['0', '1/3', '2/3', '1'])\n",
    "\n",
    "ax.get_xaxis().set_ticks_position('bottom')\n",
    "ax.get_yaxis().set_visible(False)\n",
    "ax.spines['right'].set_visible(False)\n",
    "ax.spines['left'].set_visible(False)\n",
    "ax.spines['top'].set_visible(False)\n",
    "ax.spines['bottom'].set_visible(False)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We don't need to directly find the distribution of $|X-Y|$, because we are only interested in the _average distance between points $x,y$_. Using LOTUS, we have:\n",
    "\n",
    "\\begin{align}\n",
    "  \\mathbb{E}|X-Y| &= \\int_{0}^{1} \\int_{0}^{1} |x-y| \\, dx dy &\\quad \\text{since }X,Y \\text{ are i.i.d., the joint PDF }=1 \\\\\n",
    "  &= \\iint_\\limits{x \\gt y} (x-y) \\, dx dy + \\iint_\\limits{x \\le y} (y-x) \\, dx dy &\\quad \\text{split into 2 integrals to deal with abs. value} \\\\\n",
    "  &= 2 \\int_{0}^{1} \\int_{y}^{1} (x-y) \\, dx dy  &\\quad \\text{by symmetry; and since } x \\gt y \\text{, inner integral starts from }y \\\\\n",
    "  &= 2 \\int_{0}^{1} \\left( \\frac{x^2}{2} - yx \\right) \\bigg|_{y}^{1} \\, dy \\\\\n",
    "  &= 2 \\int_{0}^{1} \\frac{1}{2} - y + \\frac{y^2}{2} \\, dy \\\\\n",
    "  &= 2 \\left( \\frac{1}{6} \\right) \\\\\n",
    "  &= \\boxed{ \\frac{1}{3} }\n",
    "\\end{align}\n",
    "\n",
    "----"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But looking at that line-segment illustration above suggests another way of looking at this problem.\n",
    "\n",
    "Let $M = max(X,Y)$\n",
    "\n",
    "Let $L = min(X,Y)$\n",
    "\n",
    "And so\n",
    "\n",
    "\\begin{align}\n",
    "  |X-Y| &= M - L \\\\\n",
    "  \\\\ \n",
    "  \\mathbb{E}(M-L) &= \\frac{1}{3} &\\quad \\text{from our previous proof} \\\\\n",
    "  \\mathbb{E}(M) - \\mathbb{E}(L) &= \\frac{1}{3} &\\quad \\text{since } X,Y \\text{ are i.i.d.} \\\\\n",
    "  \\\\\n",
    "  \\mathbb{E}(M+L)&= \\mathbb{E}(M) + \\mathbb{E}(L) &\\quad \\text{by linearity} \\\\\n",
    "  &= \\frac{1}{2} + \\frac{1}{2} &\\quad \\text{since} X \\sim \\operatorname{Unif}(0,1) \\text{, } Y \\sim \\operatorname{Unif}(0,1) \\\\\n",
    "  &= 1 \\\\\n",
    "  \\\\\n",
    "  \\mathbb{E}(M) &= \\frac{2}{3} \\\\\n",
    "  \\mathbb{E}(L) &= \\frac{1}{3} \n",
    "\\end{align}\n",
    "\n",
    "----"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Chicken-egg Problem\n",
    "\n",
    "Say we have $N$ eggs; let $N \\sim \\operatorname{Pois}(\\lambda)$\n",
    "\n",
    "Some of the eggs hatch, while some don't. Each egg hatches with probability $p$. The event of an egg hatching is independent of the others.\n",
    "\n",
    "Let $X$ be the number of eggs out of $N$ that do hatch, so $X|N \\sim \\operatorname{Bin}(N,p)$.\n",
    "\n",
    "Let $Y$ be the number of eggs that don't hatch, so $X+Y = N$.\n",
    "\n",
    "Find the joint PMF of $X,Y$. Are they independent?\n",
    "\n",
    "\\begin{align}\n",
    "  P(X{=}i, Y{=}j) &= \\sum_{n=0}^{\\infty} P(X{=}i, Y{=}j |N{=}n) \\, P(N{=}n) \\\\\n",
    "  \\\\\n",
    "  &=  P(X{=}i, Y{=}j |N{=i+j}) \\, P(N{=i+j})　&\\quad \\text{since we know that } i + j = n \\\\\n",
    "  \\\\\n",
    "  &=  P(X{=}i | N{=i+j}) \\, P(N{=i+j})　&\\quad Y{=}j \\text{ is redundant} \\\\\n",
    "  \\\\\n",
    "  &= \\binom{i+j}{i} p^i \\, q^j \\cdot \\frac{e^{-\\lambda} \\lambda^{i+j}}{(i+j)!} &\\quad \\text{from definition of binomial and Poisson} \\\\\n",
    "  \\\\\n",
    "  &=  \\frac{(i+j)!}{i! \\, j!} p^i \\, q^j \\cdot \\frac{e^{-\\lambda} \\lambda^{i+j}}{(i+j)!} \\\\\n",
    "  \\\\\n",
    "  &= \\frac{(\\lambda p)^i}{i!} \\, \\frac{(\\lambda q)^j}{j!} \\, e^{-\\lambda} \\\\\n",
    "  \\\\\n",
    "  &= \\left(e^{-\\lambda p} \\frac{(\\lambda p)^i}{i!} \\right) \\left(e^{-\\lambda q}  \\frac{(\\lambda q)^j}{j!} \\right) &\\quad \\text{since } p + q = 1 \\\\\n",
    "  \\\\\n",
    "  &\\Rightarrow X,Y \\text{ are independent, } X \\sim \\operatorname{Pois}(\\lambda p), Y \\sim \\operatorname{Pois}(\\lambda q)\n",
    "\\end{align}\n",
    "\n",
    "----"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "View [Lecture 19: Joint, Conditional, and Marginal Distributions | Statistics 110](http://bit.ly/2oRj1aU) on YouTube.d"
   ]
  }
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