# Final Exam SOLUTION

Four independent switches are arranged as follows
The switches are each closed with probabilityl/2. Evaluate the probability of a connection from a to c. [HINT: Begin by finding the probability of a connection from a to b.]

The random variables X and Y have the joint probabilities
P(X = 1,Y = 1) = 1 I 4
P(X= -2,Y = 1)= 114
P(X=O,Y=1)=114
P(X=2,Y=2)=114.
Are X and Y independent?

A game consists of independent trials with equal probabilities of success or failure. The game ends when there is a third failure. Let N be the number of trials up to and including the third failure. Evaluate the probability that the game lasts longer than four trials, that is P(N 4).

Two independent and identical players, A and B, engage in a game with probability p of success and probability q of failure. The game ends for a player with the first failure.

Denote by NA and Nathe number of plays for players A and B up to and including their first failure.

The expected number of plays for each player is 3, that is E[NA] = E[N8 ] = 3. Evaluate the probability that NA = Na.

Two manufacturers, A and B, produce hybrid car batteries. Thirty percent of the batteries are defective. Manufacturer A produces twenty percent of the batteries. Of the batteries produced by manufacturer B, twenty percent are defective. Determine the probability that a battery produced by manufacturer A is defective?

The joint density of X andY is given by
{
(x+y), O\$x\$l~ 0\$y\$l
fxy(x,y)= 0, otherwise.
Evaluate P(X Y).

The random variables X and Y have the joint density
I ( )-{ e-Y, O~x~y<oo XY X,y -
' 0 , otherwise.
Evaluate the covariance of X andY.

Let Xk, k = 1,2, ... be a collection of independent random variables with probabilities Consider the sum Evaluate the probability P(Z 2). [HINT: You may find the characteristic function a useful tool.]