Chapter 6.4, Problem 11P

Let X,Y ~ U(0,1) be iid. Derive the distribution of Z=X/Y

Suppose X1, . . . , Xn are i.i.d. from a continuous distribution with p.d.f. fθ(x) = 1/θ if 0 ≤ x ≤ θ, where θ > 0 is an unknown parameter. (a) Find E(X1) (b) Find the MME for θ. (c) Compute the variance of your MME from part (a).

Let X1, . . . , Xn be iid random variables with the shifted exponential distribution f(x | θ) = e−(x−θ),  x > θ,            =   0, x  ≤ θ(a) Find the sampling distribution of the MLE T = X(1). (b) Show that this distribution is equivalent to the distribution of T = W + θ where W ∼ Exp(n).