3. Hand in 9.24 Let Y1, Y2, Y3, ...Y, be independent standard normal random variables. a What is the distribution of , Y?? b Let W, = E, Y}. Does W, converge in probability to some constant? If so,...

Extracted text: 3. Hand in 9.24 Let Y1, Y2, Y3, ...Y, be independent standard normal random variables. a What is the distribution of , Y?? b Let W, = E, Y}. Does W, converge in probability to some constant? If so, what is the value of the constant? 4. Hand in 9.30 Let Y1, Y2, ..., Y, be independent random variables, each with probability density function [ 3y², 0<><1, f(v)="0," elsewhere.="" show="" that="" y="" converges="" in="" probability="" to="" some="" constant="" and="" find="" the="">


Extracted text: 3. Handin Ex. 9.84 A certain type of electronic component has a lifetime Y (in hours) with probability density function given by SO10) = { () ye-®, y > 0, otherwise. 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 Εί) and V (). c Suppose that 0 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?