1. GENERATING NON-UNIFORM RANDOM DEVIATES
1/1/2019
VMASC MSIM 710/810

2. INTRO
X is a random variable with non-uniform distribution F(x) = Prob[X <= x]
Inter-arrival time (exponential)
Heights of basketball players (triangular)
Target location error (2-D Normal)
Failure time (Weibull)
Coin flips before H-H-H (Geometric)
Dice Rolls (Binomial)
Drawing an “Out” (Bernoulli)
1/1/2019
VMASC MSIM 710/810

3. BASICS Generate one or more independent U[0, 1]
Create sample of X using these as input
Use few Uniforms
Be Fast
Be Simple
Be EXACT (we’ll relax this later)
1/1/2019
VMASC MSIM 710/810

4. INVERSE TRANSFORM METHOD
Generate U~U[0, 1]
Find x such that F(x) = U, return x
F(x) = P[X <= x]
Since F is a monotonically increasing function in x, we can reliably use F-1
F-1(U) = x
1/1/2019
VMASC MSIM 710/810

5. INVERSE TRANSFORM METHOD PROOF!
1/1/2019
VMASC MSIM 710/810

6. PROOF BY PICTURE
1/1/2019
VMASC MSIM 710/810

7. EXAMPLE
Weibull (a = 1.5, b = 6) example
1/1/2019
VMASC MSIM 710/810

8. WEIBULL
1/1/2019
VMASC MSIM 710/810

9. WEIBULL Trickeration: 1-U and U are identically distributed 1/1/2019
VMASC MSIM 710/810

10. SPECIAL CASE: EXPONENTIALS
Exponential = Weibull with…
b = 1/l
a = 1
X = -1/l * ln(1 – U)
1/1/2019
VMASC MSIM 710/810

11. DISCRETE DISTRIBUTIONS
P[X=xi]=pi
1/1/2019
VMASC MSIM 710/810

12. NORMALS
F, the CDF of the Normal Distribution, cannot be written down in closed form
What to Do?
exploit the Central Limit Theorem
use conditional probability for a new method
1/1/2019
VMASC MSIM 710/810

13. EXPLOITING THE CLT
Result: sum of n i.i.d. random variables (m, s2)  N(nm, ns2)
Method (Composite)
Generate U1, U2, ..., U30 ~U[0, 1]
SUM ~ N(30 * ½, 30 * 1/6)
(SUM – 15)/sqrt(5) ~ N(0, 1)
CLT approximation is more exact with data having symetric distributions
“30” comes from very old folklore (Galton)
1/1/2019
VMASC MSIM 710/810

14. ACCEPTANCE-REJECTION METHOD
To generate variate X from inaccessible CDF FX...
Generate x uniformly in the Range of X
Generate U~U[0, max(f(x))]
if U <= fX(x), return x
otherwise, try again
1/1/2019
VMASC MSIM 710/810

15. Proof:
Consider
this
slice
1/1/2019
VMASC MSIM 710/810

16. EXACT COMPOSITION METHODS
n Summed exponentials make a Gamma(n, l)
Two summed Uniforms make a Triangular
Summed Bernouli’s make a Binomial
See ... Handy T&R Facts.pdf
1/1/2019
VMASC MSIM 710/810

17. SUMMARY Inverse transform method is exact and simple
Composite methods are easy
You need to know a little about the probability you are modeling
Excel has inverse functions built in!
1/1/2019
VMASC MSIM 710/810