Problem 3: Let W,..., Wz be a random sample from an Exponential distribution with density f(w\B) = Bexp (-w) for w > 0, with >0, and let Z be a zero-truncated Poisson random variable with p.d.f. f(z|) = for z = 1,2,3,...co, with 1 > 0, and Z and W random variables are all independent. Az z!(e-1)' 1. Define X = min (W,..., Wz) and prove the conditional distribution of the random variable X, given Z = z, is an Exponential distribution with parameter z. II. Also show that the marginal probability density function of X(>0) [5] [9] f(x|A, B)= (1-e) A-B+A exp(-B) e III. Find the c.d.f. of the X, i.e. F(x|A, B). [5] III. Suppose we have n observations (X1,...,xn). Obtain the likelihood function and then find the MLE of (2,) using calculus and show that they need to be solved iteratively. [6]