Let Y1,..., Yn be random sample from a Gamma (α, ẞ), show 1/nΣ1 Y² converges in probability and find its limit.

Let X ~ Exponential(a) where a > 0 is a rate parameter. For a constant 0 > 0, show that the cumulative distribution function (CDF) of the random variable Y = Oe* is () Fy(y) = 1 – for y > 0, and zero otherwise. For what values of a is the expectation E(Y) defined?

Suppose you have the following random sample with n independent obser- vations: X1, X2,..., Xn iid * Uniform(-0,0) (a) the minimum of the sample, denoted by X(1)· Find the cumulative distribution function and probability density function of (b) What does X(1) converge in probability to? Justify your answer. What does iAi converge in probability to? Justify your answer. (c) nX(1)

Let X₁,..., Xn be an iid random sample from a Uniform(0, 0), where 0 > 0. Let X(n) = max{X₁, (a) Find the mean and variance of X(n). (b) Find the limiting distribution of Zn = n (0 – X (n)). (c) Find the limiting distribution of log(Zn). Xn}