a. Prove that Lahiri’s method results in a probability proportional to size sample with replacement. Let J be an integer with J ≥ max {Mi}. Let U1, U2, . . . be discrete uniform {1, . . . , N} random variables, let V1, V2, . . . be discrete uniform {1, . . . , J} random variables, and assume all Ui and Vj are independent. To select the first psu, we generate pairs (U1, V1), (U2, V2) …until Vj ≤ MUj .

b. Suppose the population has N psus, with sizes M1,M2, . . . ,MN. Let X represent the number of pairs of random numbers that must be generated to obtain a sample of size n. Find E [X].