1. Let X and Y be independent and suppose that each has a Uniform(0, 1) distribution. Let Z = min{X, Y }. Find the density fZ (z) for Z.
Hint:
It might be easier to ?rst ?nd P(Z > z).
2. Let X have cdf F . Find the cdf of X+ = max{0,X}.
3. Let X ∼ Exp(β). Find F (x) and F -1(q).
4. Let X and Y be independent. Show that g(X) is independent of h(Y ) where g and h are functions.
5. Suppose we toss a coin once and let p be the probability of heads. Let X denote the number of heads and let Y denote the number of tails.
(a) Prove that X and Y are dependent.
(b) Let N ∼ Poisson(λ) and suppose we toss a coin N times. Let X and Y be the number of heads and tails. Show that X and Y are independent.
6. Prove Theorem 2.33.
7. Let X ∼ N (0, 1) and let Y = eX .
(a) Find the pdf for Y . Plot it.
(b) (Computer Experiment.) Generate a vector x = (x1,..., x10,000) con- sisting of 10,000 random standard Normals. Let y = (y1,..., y10,000) where yi = exi . Draw a histogram of y and compare it to the pdf you found in part (a).