ONE

(1) How could random numbers that areuniform on the interval [0,1] be transformed into random numbers that are uniform on the interval [-11,17]?

(2) Use the linear congruential method togenerate a sequence of three two-digit random integers and corresponding random numbers. Let and .

(3) Use the multiplicative congruential method to generate a sequence of four three -digit random integers and corresponding random numbers. Let and .

(4) The sequence of numbers 0.54,0.73,0.98,0.11 and 0.68 has been generated. Use the Kolmogorov smirnov test with to learn whether the hypothesis that the numbers are uniformely distributed on the interval [0,1] can be rejected.

(5) Figure out whether these linear congruential generators can achieve a maximum period; also, state restrictions on to obtain this period.

(a)The mixed congruential method with a=2,814,749,767,109

C=59,482,661,568,307, .

(b)The multiplicative congruential generator with

(c)The mixed congruential generator with

(d)The multiplicative congruential generator with

.

(6) Test the following sequence of numbers for uniformity and independence :0.594,0.928,0.515,0.055,0.507,0.351,0.262,0.797,0.788,0.442,0.097,0.798,0.227,0.127,0.474,0.825,0.007,0.182,0.929,0.852.

(7) Lecuyer [1988] provides a generator that combines three multiplicative generators, with: a=157, m=32,363, , and .The period ofthis generator

Is approximately .Generate 5 random numbers with the combined generator, usingthe initial seeds:

and for the individual generators.

TOTAL: [70MKS]