{ "cells": [ { "cell_type": "markdown", "metadata": { "nbgrader": { "grade": false, "grade_id": "header", "locked": true, "schema_version": 1, "solution": false } }, "source": [ "Problem Set 2 Probability Algebra\n", "====\n", "\n", "Using the graph below, you have a samples space where each element is a valid path on the graph that does not include any vertex twice and must includes at least two vertices. For example, ACD and CB are valid paths but D and DCD are not.Define the random variable $X$ to be 1 when a path contains a $B$ and random variable $Y$ to be the length of the path. " ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false, "nbgrader": { "grade": false, "grade_id": "graph", "locked": true, "schema_version": 1, "solution": false } }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from IPython.display import Image\n", "Image(url='data:image/png;base64,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')" ] }, { "cell_type": "markdown", "metadata": { "nbgrader": { "grade": false, "grade_id": "2-1", "locked": true, "schema_version": 1, "solution": false } }, "source": [ "#### Problem 2.1\n", "\n", "What is the sample space? *Hint: the size is between 10 and 20*" ] }, { "cell_type": "markdown", "metadata": { "nbgrader": { "grade": true, "grade_id": "2-1-a", "locked": false, "points": 4, "schema_version": 1, "solution": true } }, "source": [ "1. AC\n", "1. ACB\n", "1. ACD\n", "1. ACBD\n", "1. BD\n", "1. BDA\n", "1. BDC\n", "1. BDAC\n", "1. CB\n", "1. CD\n", "1. CBD\n", "1. CDA\n", "1. CBDA\n", "1. DA\n", "1. DC\n", "1. DAC\n", "1. DCB\n", "1. DACB" ] }, { "cell_type": "markdown", "metadata": { "nbgrader": { "grade": false, "grade_id": "2-2", "locked": true, "schema_version": 1, "solution": false } }, "source": [ "#### Problem 2.2\n", "\n", "What is $P(X = 1)$?\n" ] }, { "cell_type": "markdown", "metadata": { "nbgrader": { "grade": true, "grade_id": "2-2-a", "locked": false, "points": 4, "schema_version": 1, "solution": true } }, "source": [ "$$\n", "\\frac{11}{18}\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "nbgrader": { "grade": false, "grade_id": "2-3", "locked": true, "schema_version": 1, "solution": false } }, "source": [ "#### Problem 2.3\n", "\n", "What is $P(Y = 3)$?" ] }, { "cell_type": "markdown", "metadata": { "nbgrader": { "grade": true, "grade_id": "2-3-a", "locked": false, "points": 4, "schema_version": 1, "solution": true } }, "source": [ "$$\n", "\\frac{8}{18}\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "nbgrader": { "grade": false, "grade_id": "2-4", "locked": true, "schema_version": 1, "solution": false } }, "source": [ "#### Problem 2.4\n", "\n", "What is $P(Y = 3 | X = 0)$?" ] }, { "cell_type": "markdown", "metadata": { "nbgrader": { "grade": true, "grade_id": "2-4-a", "locked": false, "points": 4, "schema_version": 1, "solution": true } }, "source": [ "We can reduce the sample space to paths which do not contain $B$ and then count those which have length $3$:\n", "\n", "$$\n", "\\frac{3}{7}\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "nbgrader": { "grade": false, "grade_id": "2-5", "locked": true, "schema_version": 1, "solution": false } }, "source": [ "#### Problem 2.5\n", "\n", "What is $P(Y = 3, X = 0)$?" ] }, { "cell_type": "markdown", "metadata": { "nbgrader": { "grade": true, "grade_id": "2-5-a", "locked": false, "points": 4, "schema_version": 1, "solution": true } }, "source": [ "We can just count the number of length 3 paths that do not include $B$\n", "\n", "$$\n", "\\frac{3}{18}\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "nbgrader": { "grade": false, "grade_id": "2-6", "locked": true, "schema_version": 1, "solution": false } }, "source": [ "#### Problem 2.6\n", "\n", "Are $Y$ and $X$ independent?" ] }, { "cell_type": "markdown", "metadata": { "nbgrader": { "grade": true, "grade_id": "2-6-a", "locked": false, "points": 4, "schema_version": 1, "solution": true } }, "source": [ "No, because $P(Y=3) \\neq P(Y = 3 | X = 0)$" ] }, { "cell_type": "markdown", "metadata": { "nbgrader": { "grade": false, "grade_id": "2-7", "locked": true, "schema_version": 1, "solution": false } }, "source": [ "#### Problem 2.7\n", "\n", "Define $Z$ to be 1 when a path length is greater than 2. Are $X$ and $Z$ conditionally independent on $Y$?" ] }, { "cell_type": "markdown", "metadata": { "nbgrader": { "grade": true, "grade_id": "2-7-a", "locked": false, "points": 4, "schema_version": 1, "solution": true } }, "source": [ "We are being asked if $P(X, Z | Y) = P(X | Y) P(Z | Y)$ for all possible values of the rvs. \n", "\n", "We can make a table for this expression. It will be convenient to know that $P(Y = 2) = 6 / 18$, $P(Y = 3) = 7 / 18$, $P(Y = 4) = 5 / 18$\n", "\n", "$$\n", "\\begin{array}{lcc|cr}\n", "X & Z & Y & P(X, Z | Y) & P(X | Y) P(Z | Y)\\\\\n", "\\hline\n", "0 & 0 & 2 & 2 / 6 & 2 / 6 \\times 1\\\\\n", "1 & 0 & 2 & 4 / 6 & 4 / 6 \\times 1\\\\\n", "0 & 1 & 3 & 5 / 7 & 5 / 7 \\times 1\\\\\n", "1 & 1 & 3 & 2 / 7 & 2 / 7 \\times 1\\\\\n", "0 & 1 & 4 & 0 / 5 & 0 / 5 \\times 1\\\\\n", "1 & 1 & 4 & 5 / 5 & 5 / 5 \\times 1\\\\\n", "\\end{array}\n", "$$\n", "\n", "So yes" ] }, { "cell_type": "markdown", "metadata": { "nbgrader": { "grade": false, "grade_id": "2-8", "locked": true, "schema_version": 1, "solution": false } }, "source": [ "#### Problem 2.8\n", "\n", "Are $X$ and $Y$ conditionally independent on $Z$?" ] }, { "cell_type": "markdown", "metadata": { "nbgrader": { "grade": true, "grade_id": "2-8-a", "locked": false, "points": 4, "schema_version": 1, "solution": true } }, "source": [ "No, for example\n", "\n", "$$\n", "P(X=1, Y=3 | Z=1) = \\frac{2}{7}\n", "$$\n", "\n", "$$\n", "P(X = 1 | Z = 1) P(Y = 3 | Z = 1) = \\frac{7}{12} \\frac{7}{12} \\neq \\frac{2}{7}\n", "$$" ] } ], "metadata": { "celltoolbar": "Create Assignment", "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.5.2" } }, "nbformat": 4, "nbformat_minor": 2 }