Inverse Transform Method and Random Variate Generation in Simulation Modeling

1. Lecture 25Simulation and Modeling Md. Tanvir Al Amin, Lecturer, Dept. of CSE, BUET CSE 411

2. Recap : Inverse Transform Method • Find cumulative distribution function • Let it be F • Find F-1 • X = F-1(R) is the desired variate where R ~ U(0,1) • Other methods are • Accept Reject technique • Composition method • Convolution technique • Special observations

3. Random variate for Continuous Distribution • We already covered for continuous case : • Proof of inverse transform method(Kelton 8.2.1 pg440) • Exponential distribution (Banks 8.1.1) • Uniform distribution (Banks 8.1.2) • Weibull distibution (Banks 8.1.3) • Triangular distribution (Banks 8.1.4) • Empirical continuous distribution (Banks 8.1.5) • Continuous distributions without a closed form inverse (Self study – Banks 8.1.6)

4. Inverse Transform method for Discrete Distributions Banks Chapter 8.1.7 + Kelton 8.2.1

5. Things to remember • If R ~ U(0,1), A Random variable X will assume value x whenever, • Also we should know the identities : We could use this identity also, It results in F-1(R) = floor(x) which is not useful because x is already an integer for discrete distribution.

6. Discrete Distributions F(x) = F(1) = 0.8 R1 = 0.65 F(x-1) = F(0) = 0.4 0 1 2 3 X = 1

7. Empirical Discrete Distribution • Say we want to find random variate for the following discrete distribution. • At first find the F

8. Empirical Discrete Distribution 1.0 R1 = 0.73 0.5 0 1 2 3

9. Discrete Uniform Distribution • Consider 1 2 3 … K-1 k

10. Discrete Uniform Distribution • Generate R~U(0,1) , output X= 1 If generated output X= 2 output X= 3 output X= 4 , output X= i 1 2 3 … K-1 k

11. Discrete Uniform Distribution , output X=1 If generated , output X=2 , output X=3 , output 4 , output X = i Imp : Expand this method for a general discrete uniform distribution DU(a,b)

12. Discrete Uniform Distribution • Algorithm to generate random variate for p(x)=1/k where x = 1, 2, 3, … k • Generate R~U(0,1) uniform random number • Return ceil(R*k)

13. Geometric Distribution Recall :

14. Geometric Distribution • Algorithm to generate random variate for • Generate R~U(0,1) uniform random number • Return ceil(ln(1-R)/ln(1-p)-1)

15. Proof of inverse transform method for Discrete case • Kelton 8.2.1 Page 444 • We need to show that

16. Disadvantage of Inverse Transform Method • Kelton page 445-448 • Disadvantage • Need to evaluate F-1. We may not have a closed form. In that case numerical methods necessary. Might be hard to find stopping conditions. • For a given distribution, Inverse transform method may not be the fastest way

17. Advantage ofInverse Transform Method • Straightforward method, easy to use • Variance reduction techniques has an advantage if inverse transform method is used • Facilitates generation of order statistics.

18. Composition Technique Kelton 8.2.2

19. Composition Technique • Applies when the distribution function F from which we wish to generate can be expressed as a convex combination of other distributions F1, F2, F3, … • i.e., for all x, F(x) can be written as

20. Composition Algorithm • To generate random variate for • Step 1 : Generate a positive random integer j such that • Step 2 : Return X with distribution function Fj

21. Geometric Interpretation • For a continuous random variable X with density f, we might be able to divide the area under f into regions of areas p1,p2,…(corresponding to the decomposition of f into its convex combination representation) • Then we can think of • step 1 as choosing a region • Step 2 as generating from the distribution corresponding to the chosen region

22. Example 0 a 1-a 1

23. Example

24. Proof of composition technique • Kelton page 449