#1
A random point (X,Y ) is distributed uniformly on the square [−1,1]2 where (x,y) ∈ [−1,1]2 means x ∈ [−1,1] and y ∈ [−1,1]. That is, the joint pdf is fX,Y (x,y) = 1/4 on the square. Determine the probabilities of the following events.
(a) X2 + Y 2 < 1
(b) 2X − Y > 0
(c) |X + Y | < 2
(d) |X − Y | > 1/2
(e) max(X,Y ) > 1/2
#2
Let X and Y be random variables for which the joint pdf is
fX,Y (x,y) = 2(x + y),0 ≤ x ≤ y ≤ 1.
Find the pdf of X + Y .
#3
Let X and Y be independent random variables with distribution Exp(λ) for some λ > 0. Find the pdfs of X − Y and X + Y and prove your answer.
#4
Let X ∼ Gamma(α1,β) and Y ∼ Gamma(α2,β) be independent. Let U = X/(X + Y ) and V = X + Y . (a) Find the cdfs of U and V . (b) Show that U and V are independent random variables.
#5
Let X1 and Y be independent random variables with distribution Exp(1).
Let D = X − Y and Q = X/Y .
(a) Write the joint pdf of X and D and the conditional pdf of X given D = 0.
(b) Write the joint pdf of X and Q and the conditional pdf of X given Q = 1.
#6
Suppose the distribution of Y conditioned on X = x is a normal distribution with mean x and variance x2 and that the marginal pdf of X is uniform on
(0,1).
(a) Compute E[Y ],Var[Y ], and Cov(X,Y ).
(b) Prove that Y/X and X are independent.
#7
Suppose that X and Y are random variables such that E[Y |X] = 10 − X and E . Find the correlation coefficient between X and Y .
#8
Let X and Y be random variables such that X has the standard normal distribution and the conditional distribution of Y given X is the normal distribution with mean 2X −3 and variance 12. Find the marginal pdf of Y and the correlation coefficient between X and Y .
