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)