Question

3. Consider the following pdf:

                                   Q(x) = {3x^2 / 2 -1? x ?1

                                              {0 otherwise

a. Set up an acceptance-rejection algorithm for this distribution. Use the majorizing function g(x) = 3/2.

b. Assuming a linear congruential generator with parameters a: 21, m: 100, c: 13, and x0: 7, generate two random variates from the distribution q(x).

c. Can you make any comments about the potential period of the LCG in part b?                                                                                                        

Answer 1

Let random variable X has a pdf

Let   has common support with X

a generic accept/reject algorithm to generate draws from a random variable X is given by

1. Generate a draw from and from U from uniform distribution
2. Accept X=V if , else return to step 1

We have been given the majorizing function g(x) = 3/2.. We want to find a pdf h(x) such that

Let c=3

let us get the pdf using

h(x) is a uniform distribution in the interval [-1,1]

That means we have

Finally

The steps then to generate 1 random variable from Q(x) is

1. Generate a draw from h(v) using and  
2. Accept X=V if , else return to step 1

b) Generating random numbers using LCG

with a=21, m = 100 c=13 and

we can get the following

and so on.

We convert this into a draw from uniform [0,1] by deviding by m=100, that is a draw from uniform (0,1)

To generate a draw from uniform [a,b], we use or

to generate from [-1,1] use

The following table shows the output for 10 random numbers

x
60
73
46
79
72
25
38
11
44
37
90

Following are the simulatins

1. Simulation 1
   1. X1 = 60, V= -1+2*60/100 = 0.2
   2. X2=73, U = 73/100 = 0.73
   3. Q(V)/(3/2) = v^2= 0.2^2 = 0.04
   4. 0.73<0.04? False, reject V
2. sim 2
   1. X3 = 46, V= -0.08
   2. X4=79, U = 79/100 = 0.79
   3. Q(V)/(3/2) = v^2= 0.0064
   4. U<0.0064? False, reject V

We do the following in excel

The output for 2 numbers is

0,80 and 0,52 are the first 2 numbers genaretd from f(x)

c) this period of LCG in part a is m=100 due to the following conditions

• c=13 is relatively prime to m=100
• a-1= 21-1=20 and the prime factors of m=100 are 5 and 2. 20 is divisible by both 5 and 2
• a-1 = 20 is a multiple of 4 and m=100 is a multiple of 4