# Question: Let f x y 3 2 x2 y 1

Let f(x, y) = 3/2, x2 ≤ y ≤ 1, 0 ≤ x ≤ 1, be the joint pdf of X and Y.

(a) Find P(0 ≤ X ≤ 1/2).

(b) Find P(1/2 ≤ Y ≤ 1).

(c) Find P(X ≥ 1/2, Y ≥ 1/2).

(d) Are X and Y independent? Why or why not?

(a) Find P(0 ≤ X ≤ 1/2).

(b) Find P(1/2 ≤ Y ≤ 1).

(c) Find P(X ≥ 1/2, Y ≥ 1/2).

(d) Are X and Y independent? Why or why not?

## Answer to relevant Questions

Using Example 4.4-2, (a) Determine the variances of X and Y. (b) Find P(−X ≤ Y). Let X and Y have a bivariate normal distribution with μX = 70, σ2X = 100, μY = 80, σ2Y = 169, and ρ = 5/13. Find (a) E(Y | X = 72). (b) Var(Y| X = 72). (c) P(Y ≤ 84, | X = 72). The pdf of X is f(x) = 2x, 0 < x < 1. (a) Find the cdf of X. (b) Describe how an observation of X can be simulated. (c) Simulate 10 observations of X. Let the distribution of W be F(9, 24). Find the following: (a) F0.05(9, 24). (b) F0.95(9, 24). (c) P(0.277 ≤ W ≤ 2.70). Let X and Y be independent random variables with nonzero variances. Find the correlation coefficient of W = XY and V = X in terms of the means and variances of X and Y.Post your question