1. Random-Variate Generation
Andy Wang
CIS
Computer Systems
Performance Analysis

2. Random-Variate Generation
Methods to generate nonuniform variables
Each method is applicable to a subset of the distribution
For a distribution, one method may be more efficient than the others

3. Inverse Transformation
Observation
u = CDF F(x) is uniformly distributed between 0 and 1
x can be generated via F-1(u)
A powerful technique
If F(x) and F-1(x)
can be computed

4. Empirical Inverse Transformation
Network packet sizes, f(x) =
64 bytes, 70%
128 bytes,10%
512 bytes, 20%
CDF F(x) =
0.0, 0 < x < 64
0.7, 64 < x < 128
0.8, 128 < x < 512
1.0, 512 < x

5. Empirical Inverse Transformation
F-1(u) =
64, 0 < u < 0.7
128, 0.7 < u < 0.8
512, 0.8 < u < 1

6. Inverse Transformation Example
Exponential distribution
f(x) = e-x
CDF F(x) = 1 - e-x = u
Inverse transformation
x = - ln(1-u)/ 
Given that u = U(0,1)
x = - ln(u)/ 

7. Rejection
Useful if a PDF g(x) exists so that cg(x) envelopes PDF f(x), where c is a constant
Steps
Generate x with PDF g(x)
Generate y uniform on [0, cg(x)]
If y < f(x), return x, else go to step 1

8. Rejection Example f(x) = 20x(1 - x)3 Let g(x) = U(0,1) Steps
Generate x on [0, 1] according to U(0,1)
Generate y uniform on [0, 2.058]
If y < 20x(1 – x)3
return x
else go to step 1
reject
accept

9. Composition (Decomposition)
Can be used
If CDF F(x) is a weighted sum of CDFs
Or, if PDF f(x) is a weighted sum of PDFs
Steps
Generate u1 ~ U(0,1), u2 ~ U(0,1)
Use u1 to choose fi(x) or Fi(x), return F-1(u2)

10. Composition Example Laplace distribution
a = 2, x > 0 with 50% probability
Steps
Generate
u1 ~ U(0,1)
u2 ~ U(0,1)
2. If u1 < 0.5
return x = -aln(u2)
else
return x = aln(u2)

11. Convolution Random variable x = y1 + y2 + … yn
x can be generated by summing n random variate yis
Example
y1 = outcome of die 1 (uniform distribution)
y2 = outcome of die 2 (uniform distribution)
x = sum of outcomes of two dice
= y1 + y2 (triangular distribution)

12. Choosing Random-Variate Generation Techniques
Use inversion if CDF is invertible
Use composition if CDF/PDF sum of other CDFs/PDFs
Use convolution if the variate a sum of other variates
Use rejection if a bounding PDF function exists
Use empirical inversion as needed

13. White Slide