1. LECTURE 25 SIMULATION 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. Banks Chapter 8.1.7 + Kelton 8.2.1 Inverse Transform method for Discrete Distributions

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 0123 R 1 = 0.65 F(x) = F(1) = 0.8 F(x-1) = F(0) = 0.4 X = 1

7. Empirical Discrete Distribution  Say we want to find random variate for the following discrete distribution.  At first find the F xp(x) 00.50 10.30 20.20 xp(x)F(x) 00.50 10.300.80 20.201.00

8. Empirical Discrete Distribution 0123 0.5 1.0 R 1 = 0.73

9. Discrete Uniform Distribution  Consider 123…K-1k

10. Discrete Uniform Distribution  Generate R~U(0,1) 123…K-1k, output X= 1 output X= 2 output X= 3, output X= i output X= 4 If generated

11. Discrete Uniform Distribution, output X=1, output X = i, output 4, output X=2, output X=3 If generated 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 of Inverse 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. Kelton 8.2.2 Composition Technique

19.  Applies when the distribution function F from which we wish to generate can be expressed as a convex combination of other distributions F 1, F 2, F 3, …  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 F j

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