1. GENERATING NON-UNIFORM RANDOM DEVIATES
12/9/2018
VMASC MSIM 710/810

2. BASICS (as evolved) Generate one or more independent U[0, 1]
Create deviate using these as input
Use few Uniforms
Be Fast
Be Simple
Be EXACT (we’ll relax this later)
12/9/2018
VMASC MSIM 710/810

3. INVERSE TRANSFORM METHOD
Generate U~U[0, 1]
Find x such that F(x) = U, return x
Recall F(x) = P[X <= x]
Since F is a monotonically increasing function in x, we can reliably use F-1
F-1(U) = x
12/9/2018
VMASC MSIM 710/810

4. INVERSE TRANSFORM METHOD PROOF!
12/9/2018
VMASC MSIM 710/810

5. PROOF BY PICTURE
12/9/2018
VMASC MSIM 710/810

6. EXAMPLE
Weibull (a = 1.5, b = 6) example
12/9/2018
VMASC MSIM 710/810

7. WEIBULL
12/9/2018
VMASC MSIM 710/810

8. WEIBULL Trickeration: 1-U and U are identically distributed 12/9/2018
VMASC MSIM 710/810

9. DISCRETE DISTRIBUTIONS
P[X=xi]=pi
12/9/2018
VMASC MSIM 710/810

10. 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
12/9/2018
VMASC MSIM 710/810

11. 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)
12/9/2018
VMASC MSIM 710/810

12. 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
12/9/2018
VMASC MSIM 710/810

13. Proof:
Consider
this
slice
12/9/2018
VMASC MSIM 710/810

14. 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
12/9/2018
VMASC MSIM 710/810