Question

1. Consider the following function

F(x) = {2x / 25 0<x<5

           {0 otherwise

a) Prove that f(x) is a valid probability function.

b) Develop an inverse-transformation for this function.

c) Assume a multiplicative congruential random number generator with parameters:

a: 23, m: 100, and xo: 17. Generate two random variates from the function for (x).

Answer 1

(a)

Given function:

   for 0 < x < 25

f(x) is a valid probability function if the Total Probability = 1.

i.e.,

between the limits 0 to 5.

Applying limits, we get:

This proves that f(x) is a valid probabilify function.

(b) Cumulative Distribution Function F(x) is got by integrating f(x) from 0 to x as follows:

between the limits 0 to x. Applying limits, we get:

The Cumulative Distribution is Uniform Distribution in (0,1).

Thus, equating F(x) to r, a random bumber generated by computer with uniform distribution in (0,1), we get:

So,

x = 5

This is the inverse-transformation for this function.

(c)

The Multiplicative Congruential Generator (MCG) is given by:

Given:

n = 0

a = 23

m = 100

Substituting, we get:

= 391 mod 100 = 91

So,

the two random numbers generated are:

91, 93