Probability and Statistics Problems and Solutions

Are X and Y independent random variables?

Solution: First we calculate the marginal pdfs

Note that fX,Y (x, y) 6= fX(x) × fY (y). So they are not independent.

What is the conditional distribution of X, given Y = .50? What is the probability that less than 50% of the initiative 1 surveys are returned, given that 50% of the initiative 2 surveys are returned?

(c) Calculate E(X/Y = y). What is the expected proportion of initiative 1 surveys returned, given that 50% of the initiative 2 surveys are returned?

Three prisoners are informed by their jailer that one of them has been 1/3 1/2

First we label the prisoners as P1, P2 and P3. Suppose P1 asks the jailer to tell him privately whether P2 or P3 will be set free. Now we define the following events.

Ai = {Pi is executed } for any i ∈ {1, 2, 3}

Note that P(Ai) = 1/3

Now suppose the jailer agrees to P1’s request. Now we consider the following cases.
Case 1: P1 is executed In this case, the jailer is equally likely to say that P2 will be set free or P3 will be set free.
Case 2: P2 is executed In this case, the jailer will have to say that P3 will be set free.
Case 3: P3 is executed In this case, the jailer will have to say that P2 will be set free.

Next we define an event B as B = { The jailer says that P2 is free}. Now we calculate P(A1|B).

Note that P(B|A1) = 1/2, P(B|A2) = 0 and P(B|A3) = 1. So we have

Let X1 and X2 be two independent random variables following uniform distribution over the interval [0, 1]. So their density functions are for any i ∈ {1, 2}. Calculate E(max{X1, X2}) and E(min{X1, X2}).

Ans. Let Z1 = max{X1, X2} and Z2 = min{X1, X2}. First we calculate the cdfs of Z1 and Z2.

Your friend tells you that he had two job interviews last week. He says that based on how the interviews went, he thinks he has a 20% chance of receiving an offer from each of the companies he interviewed with. Nevertheless, since he interviewed with two companies, he is 50% sure that he will receive at least one offer. Is he right?

Let Ai denote the event that your friend receives an offer from company i (i ∈ {1, 2}). Then

So P(A1 ∪ A2) ≤ P(A1) + P(A2) = .4. So he is wrong.

Let X be a random variable, which follows uniform distribution over the interval [−1, 1]. So pdf of f is given by

Let Y be another random variable defined as Y = X² . In class, we have shown that ρ(X, Y ) = 0. Show that X and Y are not independent.

Solution: Note that P(X ≤ −0.5) = 0.25 and P(Y ≤ .2) = P(− √0.2 ≤ X ≤ √0.2) = √0.2, but P(X ≤ −0.5, Y ≤ .2) = 0.