1. Random number generation
Chapter 3
Random number generation

2. Topics Solving problems using random number tables
Generation of random numbers that are
Discrete and uniform
Non uniform and discrete
Non uniform and continuous
Exponential
Uniform in (a,b)
Normal
Rejection method

3. Discrete and uniform numbers
Suppose the range is 1 to 5
Using table of uniform numbers we can generate the numbers as follows:
Use numbers in the table and multiply each by 4 and add 1 and truncate (or round)
The first few numbers are , , , ,
When you do the operations above the numbers generated numbers are 1,2,1,4,1
Generate 25 numbers and make a frequency distribution

4. Problems
Assuming a normal 365 day year generate first 10 dates of the year
Solution
Generate month using one column
Date using another
Why do like this?

5. Problems
The probability that a batter swings the bat is 0.7, strikes the ball is 0.6 and he is caught is Compute the number of plays that he will be out in first 10 plays
Use columns 1, 2 and 3 for the three events
The theoretical probability that he is out is 0.7 x 0.6 x 0.5 = 0.21
The expected number of plays he will be out in 10 balls is 0.21 x 10 = 2

6. Non uniform discrete distribution
Assume a single item inventory simulation in a retail store. Let us find the average sales in 10 days. You are given the following information regarding the demand of the item (let us say soap). Assume that reorder of 15 soaps and reorder point to be 5 soaps and lead time is 1 day
Suppose that the distribution of customers asking for soap in a day is
No. of customers
Probability
Cumulative probability

7. Non uniform discrete distribution
Suppose that the distribution of demand of soaps is
No. of soaps
Probability
Cum Prob
Assume initial stock of 12 soaps
Day I S R N No. customers R Ns Soaps sold F S
…..

8. How does this kind of generation work?x
P(x)
Cum P(x) and Plot of cdf:
o o

9. How does this kind of generation work?
1 is generated if the uniform random number is between 0.2 to 0.45
If 100 numbers are generated then number of times 1 is generated is ( ) x 100 = 25
There is a match between theoretical frequency and the no of numbers generated by the generator

10. Non uniform continuous distributionx
P(x) CP
For the random number in the table the interval is chosen
Use the cdf (line) in this interval and find the number corresponding to

11. Non uniform continuous distributionx
P(x) CP
o o

12. Non uniform continuous distributionx
P(x) CP
The cdf line in the interval is join of (20, 0.08) and (40, 0.24)
For the cumulative probability the random number generated is
{[( )/ ( )]*(40-20) }+20

13. Exponential Many arrivals to a queue are Poisson or pure random
The inter arrivals are exponential
Service times are often exponential
T be mean inter arrival time
 be mean inter arrival rate
Then T=1/ 
Example: Mean inter arrival time be 20 sec
Then mean arrival rate is 1 in 20 or 1/20 arrivals in a sec

14. Exponential PDF is f(x)=e- x , x≥0 CDF is y=F(x)=1- e- x , x≥0
x=- log(1-y)/ , where y is uniform in (0,1)
x=- log(y)/ 
Inverse transformation method

15. Principle of inverse transformation
F(x) y x x

16. Algorithm for exponential random number generation
Algorithm exp(lambda) Global u, x x= –ln(y)/lambda Return x End algorithm exp

17. Normal (Gaussian) PDF is not integrable No cdf function Method 1
Generate k unit uniform numbers
Find their sum
n=(sum – k/2 )/sqrt(k/12) is standard normal
If random variates that are normal with mean a and sd b are to be generated then it is n*b+a

18. Normal (Gaussian)
Convenient and quick formula for standard normal generation
Generate 12 unit uniform numbers
Find their sum
n=(sum – 6 ) is standard normal
If random variates that are normal with mean a and sd b are to be generated then it is n*b+a

19. Normal (Gaussian) Method 2 n=((-2lnx)1/2 ) cos(2y) is standard normal
Here x, y are unit uniform numbers
n=((-2lnx)1/2 ) sin(2y) is standard normal
If random variates that are normal with mean a and sd b are to be generated then it is n*b+a

20. Rejection method For obtaining samples from non uniform distribution
Basically generate uniform random numbers and accept those that meet some criteria
The criteria is designed so that generated numbers are from the given distribution

21. Rejection method f(x) q q c A p p B x
Range of p is (A,B) and that of q is (0,c)

22. Lab problems
Generate exponential, uniform in (a,b) and normal variates and plot frequencies
Generate empirical discrete and continuous distribution random variates and plot frequencies
Simulate the soap problem

23. Summary and references
Solving problems using random number tables – Chapter 6, G. Gordon
Generation of random numbers that are
Discrete and uniform - G. Gordon, Chapter 6; N.Deo, Chapter 3
Non uniform and discrete - G. Gordon, Chapter 6; N.Deo, Chapter 3
Non uniform and continuous - G. Gordon, Chapter 6; N.Deo, Chapter 3
Exponential - G. Gordon, Chapter 7; N.Deo, Chapter 3
Uniform in (a,b) - G. Gordon, Chapter 7; N.Deo, Chapter 3
Normal - G. Gordon, Chapter 7; N.Deo, Chapter 3
Rejection method - N.Deo, Chapter 3