HW 2. Math 504. Spring 2008. CSUF

by Nasser Abbasi

Problem description

This problem is a follow up on the problem described in HW1.

In this problem we are asked to derive analytically the PDF of the random variable $Y$ by conditioning on the box selected.

$Y$ is the random variable which is the observation from the following experiment: Generate random variable $X$ from uniform $\left[ 0,1\right] $. Put this number is box labeled $S$ and put twice this number in a box labeled $\bar{S}$. Next, we pick one of these 2 boxes by random. If the number inside the box selected is found to be greater than 1, then we switch the boxes and pick the number inside the second box. The random variable $Y$ is the final number selected.

Solution

We first note the following known probabilities in this problem. The probability of picking box $S$ or box $\bar{S}$ is $\frac{1}{2}$. Once we pick box $S$, then we have to switch the box. If we pick the $\bar{S}$ box, then we switch only if the observed $X$ is less than 1.

To help solve this problem, we start by drawing the decision tree describing the possible flow and assign a probability to each branch. At the end of each branch we draw the PDF of $Y$ resulting from traversing that branch only. Next, we combine (add algebraically) all the PDF's together after we scale each PDF by the probabilities found along the edges which lead to the end of the branch.


tree.png

Using the above diagram as a guide, we now calculate the PDF for $Y$ as follows (starting from the right most branch to the left most branch)

MATH