Problem 3: Let WI, ..., Wz be a random sample from an Exponential distribution with density f(w|B) = Bexp (-wB) for w > 0, with B >0, and let Z be a zero-truncated Poisson random variable with p.d.f. f(z|1) = ((ed-1)' for z = 1,2,3, ...co, with 1 > 0, and Z and W random variables are all independent. 1. Define X = min (W1, ..., Wz) and prove the conditional distribution of the random variable X, given Z = z, is an Exponential distribution with parameter Bz. [5] II. Also show that the marginal probability density function of X(> 0) [9] f ( = | X , B ) XB (1 - edje A Bx X exp(-Bx) Ill. Find the c.d.f. of the X, i.e. F(x|1, B). [5] Ill. Suppose we have n observations (X1, ..., Xn). Obtain the likelihood function and then find the MLE of (2, B) using calculus and show that they need to be solved iteratively. [6]