3. A certain type of electronic component has a lifetime Y (in hours) with probability density function given by f(x; 8) = )xe, x>0 хе elsewhere That is, Y has a gamma distribution with parameters a = 2 and 0. Let ô denote the MLE of 0. Suppose that three such components, tested independently, had lifetimes of 120, 130, and 128 hours. a. Find the MLE of 0. b. Find E(Ô) and V (ô) c. Suppose that 0 actually equals 130. Give an approximate bound that you might expect for the error of estimation.

9.84 A certain type of electronic component has a lifetime Y (in hours) with probability density function given by f(y|0)= { (1) y> 0, otherwise. That is, y has a gamma distribution with parameters α = 2 and 0. Let Ô denote the MLE of 8. Suppose that three such components, tested independently, had lifetimes of 120, 130, and 128 hours. a Find the MLE of 0. b Find E() and V (8). c 82 ye-3/0 Suppose that actually equals 130. Give an approximate bound that you might expect for the error of estimation. d What is the MLE for the variance of Y?

5.4.6 WP Determine the covariance and correlation for the joint probability density function fry(x, y) = e* over the range 0< x and 0 < y.

6.106 If Y is a continuous random variable with distribution function F(y), find the probability density function of U = F(Y).