Practice Problems for Conditional Distributions, Part 1

The following are practice problems on conditional distributions. The thought process of how to work with these practice problems can be found in the blog post Conditionals Distribution, Part 1.

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Description of Problems

Suppose X and Y are independent binomial distributions with the following parameters.

    For X, number of trials n=5, success probability \displaystyle p=\frac{1}{2}

    For Y, number of trials n=5, success probability \displaystyle p=\frac{3}{4}

We can think of these random variables as the results of two students taking a multiple choice test with 5 questions. For example, let X be the number of correct answers for one student and Y be the number of correct answers for the other student. For the practice problems below, passing the test means having 3 or more correct answers.

Suppose we have some new information about the results of the test. The problems below are to derive the conditional distributions of X or Y based on the new information and to compare the conditional distributions with the unconditional distributions.

Practice Problem 1

  • New information: X<Y.
  • Derive the conditional distribution for X \lvert X<Y.
  • Derive the conditional distribution for Y \lvert X<Y.
  • Compare these conditional distributions with the unconditional ones with respect to mean and probability of passing.
  • What is the effect of the new information on the test performance of each of the students?
  • Explain why the new information has the effect on the test performance?

Practice Problem 2

  • New information: X>Y.
  • Derive the conditional distribution for X \lvert X>Y.
  • Derive the conditional distribution for Y \lvert X>Y.
  • Compare these conditional distributions with the unconditional ones with respect to mean and probability of passing.
  • What is the effect of the new information on the test performance of each of the students?
  • Explain why the new information has the effect on the test performance?

Practice Problem 3

  • New information: Y=X+1.
  • Derive the conditional distribution for X \lvert Y=X+1.
  • Derive the conditional distribution for Y \lvert Y=X+1.
  • Compare these conditional distributions with the unconditional ones with respect to mean and probability of passing.
  • What is the effect of the new information on the test performance of each of the students?
  • Explain why the new information has the effect on the test performance?

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Partial Answers

To let you know that you are on the right track, the conditional distributions are given below.

The thought process of how to work with these practice problems can be found in the blog post Conditional Distributions, Part 1.

Practice Problem 1

    \displaystyle P(X=0 \lvert X<Y)=\frac{1023}{22938}=0.0446

    \displaystyle P(X=1 \lvert X<Y)=\frac{5040}{22938}=0.2197

    \displaystyle P(X=2 \lvert X<Y)=\frac{9180}{22938}=0.4

    \displaystyle P(X=3 \lvert X<Y)=\frac{6480}{22938}=0.2825

    \displaystyle P(X=4 \lvert X<Y)=\frac{1215}{22938}=0.053

    ____________________

    \displaystyle P(Y=1 \lvert X<Y)=\frac{10}{22933}=0.0004

    \displaystyle P(Y=2 \lvert X<Y)=\frac{540}{22933}=0.0235

    \displaystyle P(Y=3 \lvert X<Y)=\frac{4320}{22933}=0.188

    \displaystyle P(Y=4 \lvert X<Y)=\frac{10530}{22933}=0.459

    \displaystyle P(Y=5 \lvert X<Y)=\frac{7533}{22933}=0.328

Practice Problem 2

    \displaystyle P(X=1 \lvert X>Y)=\frac{5}{3386}=0.0013

    \displaystyle P(X=2 \lvert X>Y)=\frac{160}{3386}=0.04

    \displaystyle P(X=3 \lvert X>Y)=\frac{1060}{3386}=0.2728

    \displaystyle P(X=4 \lvert X>Y)=\frac{1880}{3386}=0.4838

    \displaystyle P(X=5 \lvert X>Y)=\frac{781}{3386}=0.2

    ____________________

    \displaystyle P(Y=0 \lvert X>Y)=\frac{31}{3386}=0.008

    \displaystyle P(Y=1 \lvert X>Y)=\frac{390}{3386}=0.1

    \displaystyle P(Y=2 \lvert X>Y)=\frac{1440}{3386}=0.37

    \displaystyle P(Y=3 \lvert X>Y)=\frac{1620}{3386}=0.417

    \displaystyle P(Y=4 \lvert X>Y)=\frac{405}{3386}=0.104

Practice Problem 3

    \displaystyle P(X=0 \lvert Y=X+1)=\frac{15}{8430}=0.002

    \displaystyle P(X=1 \lvert Y=X+1)=\frac{450}{8430}=0.053

    \displaystyle P(X=2 \lvert Y=X+1)=\frac{2700}{8430}=0.32

    \displaystyle P(X=3 \lvert Y=X+1)=\frac{4050}{8430}=0.48

    \displaystyle P(X=4 \lvert Y=X+1)=\frac{1215}{8430}=0.144

    ____________________

    \displaystyle P(Y=1 \lvert Y=X+1)=\frac{15}{8430}=0.002

    \displaystyle P(Y=2 \lvert Y=X+1)=\frac{450}{8430}=0.053

    \displaystyle P(Y=3 \lvert Y=X+1)=\frac{2700}{8430}=0.32

    \displaystyle P(Y=4 \lvert Y=X+1)=\frac{4050}{8430}=0.48

    \displaystyle P(Y=5 \lvert Y=X+1)=\frac{1215}{8430}=0.144

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\copyright \ 2013 \text{ by Dan Ma}

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