Cumulative_Distribution_Functions.ipynb

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    "<h1>Common Ground</h1>\n",
    "Much of the treatment of random variables (RV) makes a distinction between **discrete** and **continous** random variables. For example, discrete RVs are associated with a probability **mass** function (PMF) that is a function from a discrete value, i.e. a possible outcome, and the probability that the RV takes that outcome. Continuous RVs are associated with a probability **density** function (PDF) that is a function from any value on the real line to the probability **density** of the RV at that value (thought of as the density per unit length). \n",
    "\n",
    "The **cumulative distribution function** (CDF) is a single concept that applies to all RVs. The CDF of a RV *X*, denoted $F_X\\hspace{1pt}$ defines the probability $P\\left(X\\le x\\right)\\hspace{1pt}$\n",
    "\n",
    "$$F_X\\left(x\\right) = P\\left(X\\le x\\right) =  \\left\\{ \\begin{eqnarray} \\sum_{k\\le x} p_X\\left(k\\right) &,& \\quad \\mbox{if}\\hspace{3pt} X \\hspace{2pt} \\mbox{is discrete} \\cr\n",
    "\\int_{-\\infty}^x f_X\\left(t\\right)dt&,& \\quad \\mbox{if}\\hspace{3pt} X \\hspace{2pt} \\mbox{is continuous} \\end{eqnarray}\\right.$$"
   ]
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   "source": [
    "<h2>Properties of the CDF</h2>\n",
    "\n",
    "If *X* is discrete and takes integer values, the PMF and the CDF can be obtained as\n",
    "\n",
    "$$F_X\\left(k\\right) = \\sum_{i=-\\infty}^k p_X\\left(i\\right)$$\n",
    "\n",
    "$$p_X\\left(k\\right) = P\\left(X\\le k\\right) - P\\left(X \\le k-1\\right) = F_X\\left(k\\right) - F_X\\left(k-1\\right)$$\n",
    "\n",
    "for all integers *k*.\n",
    "\n",
    "If *X* is a continuous RV, the PDF and CDF can be obtained as\n",
    "\n",
    "$$F_X\\left(x\\right) = \\int_{-\\infty}^x f_X\\left(t\\right)dt$$\n",
    "\n",
    "$$f_X\\left(x\\right) = \\frac{dF_X}{dx}\\left(x\\right)$$"
   ]
  },
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   "source": [
    "<h3>Example: PMF/PDF For Maximum of Several Random Variables</h3>\n",
    "\n",
    "Consider the set of random variables $\\mathbf{X}=\\left\\{X_1,\\ldots,X_N\\hspace{1pt}\\right\\}$ where the random variables in **X** *are independent*. Let *Z* be the random variable equal to the maximum value over the set **X** for a given observation of **X**. We can obtain the PMF or PDF of **Z** by first considering the CDF\n",
    "\n",
    "$$\\begin{eqnarray}F_Z\\left(z\\right) &=& P\\left(X_1\\le z\\right)\\ldots P\\left(X_N \\le z\\right) \\cr\n",
    "&=& F_{X_1}\\left(z\\right)\\ldots F_{X_N}\\left(z\\right)\\end{eqnarray}$$\n",
    "\n",
    "By the properties of the CDF, the PMF or PDF can be obtained. \n",
    "\n",
    "Consider the case where **X** consists of two *independent Normal random variables* with means $\\mu_i = 0, 3\\hspace{2pt}$ respectively and variances $\\sigma^2 = 1, 16\\hspace{2pt}$ respectively and let **Z** be the maximum of **X**. \n",
    "\n",
    "Recall that for an arbitrary *Gaussian* random variable *X* with mean $\\mu\\hspace{1pt}$ and standard deviation $\\sigma\\hspace{1pt}$ we have\n",
    "\n",
    "$$F_X\\left(x\\right) = \\Phi \\left(\\frac{x-\\mu}{\\sigma}\\right)$$\n",
    "\n",
    "where \n",
    "\n",
    "$$\\Phi\\left(x\\right) = \\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^x e^{-t^2/2}dt$$\n",
    "\n",
    "is the CDF of the *standard Normal*, that is a Gausian RV with zero mean and unit variance. Thus CDF of Z is\n",
    "\n",
    "$$F_Z\\left(z\\right) = \\Phi\\left(z\\right) \\cdot \\Phi\\left(\\frac{z-3}{4}\\right)$$\n",
    "\n",
    "Now we can obtain the PDF of **Z** by applying the [Fundamental Thoerem of Calculus](http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus) and the [chain rule](http://en.wikipedia.org/wiki/Chain_rule) to obtain\n",
    "\n",
    "$$\\begin{eqnarray}f_Z\\left(z\\right) &=& \\Phi\\left(z\\right) \\cdot \\frac{d}{dz}\\Phi\\left(\\frac{z-3}{4}\\right) + \\Phi\\left(\\frac{z-3}{4}\\right) \\cdot \\frac{d}{dz}\\Phi\\left(z\\right) \\cr\n",
    "&=& \\Phi\\left(z\\right)\\cdot \\frac{1}{4}\\cdot f_Z\\left(\\frac{z-3}{4}\\right) +\\Phi\\left(\\frac{z-3}{4}\\right)\\cdot f_Z\\left(z\\right) \n",
    "\\end{eqnarray}$$\n",
    "\n",
    "where\n",
    "\n",
    "$$f_Z\\left(z\\right) = \\frac{1}{\\sqrt{2\\pi}} e^{-z^2/2}$$\n",
    "\n",
    "Below we attempt to confirm (technically we are only \"not contradicting\") this result via simulation. We will take *N* observations of **Z** for some *large N* and determine if the observed distribution *closely matched* the predicted one. "
   ]
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   "execution_count": 32,
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import scipy.stats\n",
    "numSamples = 20000\n",
    "\n",
    "def fZ(z):\n",
    "    rv = scipy.stats.norm(loc=0, scale=1) #standard normal\n",
    "    z1 = (z-3)/(4.0)\n",
    "    return rv.cdf(z)*rv.pdf(z1)/(4.0) + rv.cdf(z1)*rv.pdf(z)\n",
    "\n",
    "raynge = [(-3 + i/10.0) for i in range(180)]\n",
    "predicted = [fZ(z) for z in raynge]\n",
    "n1 = scipy.stats.norm(loc=0, scale=1)\n",
    "n2 = scipy.stats.norm(loc=3, scale =4)\n",
    "\n",
    "X1 = n1.rvs(size=numSamples)\n",
    "X2 = n2.rvs(size=numSamples)\n",
    "Z = [max(X1[i],X2[i]) for i in range(numSamples)]\n",
    "\n",
    "fig = pyplot.figure()\n",
    "ax = fig.add_subplot(111)\n",
    "\n",
    "n, bins, patches = ax.hist(Z, 100, normed=1, facecolor='green', alpha=0.75)\n",
    "ax.plot(raynge,predicted,'r', linewidth=3)\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": []
  }
 ],
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   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
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