HW 2. Math 504. Spring 2008. CSUF
by Nasser Abbasi
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 by conditioning on the box selected.
is the random variable which is the observation from the following experiment: Generate random variable from uniform . Put this number is box labeled and put twice this number in a box labeled . 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 is the final number selected.
We first note the following known probabilities in this problem. The probability of picking box or box is . Once we pick box , then we have to switch the box. If we pick the box, then we switch only if the observed 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 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.
Using the above diagram as a guide, we now calculate the PDF for as follows (starting from the right most branch to the left most branch)