probability6

Math 464 - Fall 13 - Homework 7

1. (This problem will count for twice an many points as the others since it has so many parts.) Random variables X and Y have joint density

f(x, y) = {

c exp(−x − y) if x ≥ 0, y ≥ x 0 otherwise

Note carefully the region where the density is non-zero. It is not x ≥ 0, y ≥ 0. (a) Find the constant c. (b) Compute P(Y ≥ X + 1). (c) Find the marginal densities of X and Y . (d) Are X and Y independent? (e) Compute E[XY ].

2.

f(x, y) =















1

π exp(−1

2 (x2 + y2)) if x ≥ 0, y ≥ 0

1

π exp(−1

2 (x2 + y2)) if x < 0, y < 0

0 if x ≥ 0, y < 0 0 if x < 0, y ≥ 0

(a) Compute the marginal densities fX(x) and fY (y). You can assume the fact

∫

∞

−∞

e −z2/2

dz = √ 2π

(b) Are X and Y independent?

3. Let X and Y be independent random variables. X is uniform on [0, 1] and Y is uniform on [0, 2]. Let Z = X + Y . Find the density of Z. Check your answer by computing the mean of the density you get. Hint: first compute the cdf of Z.

4. Let X and Y be independent random variables. X has an exponential distribution with parameter λ1 and Y has an exponential distribution with parameter λ2. (a) Let t > 0. Find P(min{X, Y } ≥ t) where min{X, Y } denotes the smaller of X and Y . Hint: The event min{X, Y } ≥ t is the same as X ≥ t, Y ≥ t. (b) Let Z = min{X, Y }. Use part (a) to find the cdf and then the pdf of the random variable Z. You should discover something nice.

1

5. Let X and Y be independent random variables. X has an exponential distribution with parameter λ = 1 and Y has an exponential distribution with parameter λ = 2. Let Z = min{X, Y } as in the previous problem. Compute by simulation using R or any software the quantities (a) P(Z ≤ 3) (b) E[Z]

Note that problem 5 gives exact values for these two quantities. So you can (and should) check the results of your simulation against your answer to 5. However, you should not use the result of problem 5 to do the simulation. Your simulation should generate samples of X and Y and then take the minimum to get a sample of Z.

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