A prepare to do this by generating 16 random integer

Question

You need to generate 10 random observations from the probability distribution

(a) Prepare to do this by generating 16 random integer numbers from the mixed congruential generator, x_n+1 ≡ (5x_n + 3) (modulo 16) and x₀ = 1.

(b) Use the single-digit random integer numbers from part (a) to generate the desired random observations.

(c) Note that once a particular value of X is generated in part (b), it can never be repeated because each of the 16 possible random integer numbers is generated exactly once in part (a). In which ways does this violate the desirable properties of random observations? What change would you make in what was done in parts (a) and (b) to alleviate this problem?

(d) Now convert the first 10 random integer numbers from part (a) to (approximate) uniform random numbers, and then apply the inverse transformation method to obtain the desired random observations.

(e) Does the procedure prescribed in part (d) actually give a probability of 1/10 of generating each of the 10 possible values of X each time? Explain. What change would you make in what was done in parts (a) and (d) to alleviate this problem?