A Summary of Random Variable Simulation

Ideas for Today and Tomorrow

Uniform

X is uniformly distributed on the interval [a,b]

We write X~unif(a,b)

Uses the basis for generating all random

variables can be used as a model for a quantity that is known to vary between a and b for which little else is known

Uniform

method of generation

use a random number generator included in software or write your own generator to generate Y~unif(0,1)

set X=(b-a)Y+a

Normal

X is normally distributed with mean and variance

Uses model errors in various processes

quantities that are sums of lots of other quantities

2

We write X~N( , ) 2

Normal

method of generation

generate Y~N(0,1)

• Box-Muller method

• Polar-Marsaglia method

setμσYX

Let U1 and U2 be independent unif(0,1) rv’s.

The Polar-Marsaglia Method

Let V1=2U1-1 and V2=2U2-1.

If , let1VV:S 22

21

ln(S) S2-

C

Then X1=CV1 and X2=CV2 are independent and normally distributed with mean 0 and variance 1.

Exponential

Uses lifetimes

We write X~exp(rate= )

waiting times

service times

interarrival times

X is exponentially distributed with rate

Exponential

method of generation

The inverse cdf method:

invert the cdf

x)ln(1λ1

(x)F 1

set X=F-1(U) where U~unif(0,1)

Double Exponential

X has a bilateral (double) exponential distribution with location parameter and shape parameter

as a “jump process” in finance

we write X~DE( , )

method of generation the pdf is x- ,βe

21

f(x) |x|β

consider the case 0

Double Exponential

this is a “back-to-back” exponential with rate• simulate Y~exp(rate= )

• flip a fair coin to add

shift X=Y+

Gamma

X has the gamma distribution with shape parameter and scale parameter

Uses

sum of exponential event times

time to complete a task consisting of consecutive exponential events

We write ),( ~X

Gamma

method of generation

the pdf is 0 x,exβ)Γ(

1f(x) βx1 αα

α

use accept-reject sampling to generate

,1)(Y

setY/X

Weibull X has the Weibull distribution with shape

parameter and scale parameter

Uses

time to complete a task

time to equipment failure

We write ),( Weibull~X

differs from exponential in that failure probability can vary over time

used in reliability testing

Weibull method of generation

the pdf is

x ,-x

-

e -x

f(x)1

set X=F-1(U)

the cdf is

-x

-

e-1 F(x)

invert x)]-[-ln(1 (x)F 1/-1

Beta X has the beta distribution with parameters

and1 2

well represents bounded rv’s with various kinds of skew (many shapes!)

distribution of random proportions

rough model in the absence of data

We write ), 21 beta(~X

Uses

Beta method of generation

the pdf is 1x0 ,x)-(1 x),(

1 f(x) 21

21

11

B

generate ),( ~ Xand ),( ~X 21 11 21

independently

set21

1

XXX

X

Pareto

X has the Pareto distribution with parameter

modeling stock price returns

modeling incomes

)Pareto(~X we write

Uses

monitoring production processes

Pareto

method of generation

0 x,x)(1

f(x) 1

the pdf is

set X=F-1(U)

the cdf is x)(1-1 F(x)

1

invert 1-x-1

1 (x)F 1-

/1

Cauchy

X has the Cauchy distribution with location parameter and scale parameter

mostly interesting for theoretical reasons

),Cauchy(~X we write

Uses

Cauchy

method of generation

simulate Y~Cauchy(0,1) by inverse cdf method

x- ,-x

1

1

1 f(x) 2

the pdf is

let YX

logistic X has the logistic distribution with location parameter

and scale parameter

growth models

),logistic(~X we write

Uses

logistic regression

logistic method of generation

set X=F-1(U)

x, ee 1

f(x)-x

--x

-

2

1 the pdf is

the cdf is e F(x)

-x-

1

1

invert: y1

ln - (x)F 1-

1

Gumbel

X has the Gumbel distribution with location parameter and scale parameter

modeling extreme events

),(~X we write

Uses

is the natural log of a Weibull with 1

Gumbel

method of generation

invert the cdf

x,eexp F(x)-x

-

or, take the natural log of a Weibull generated with 1

Log-Normal

We write X~LN( , )

model quantities that are products of a large number of random quantities

2

Note: and are not the mean and variance!

2

ln(X) ~ N( , ) 2 Uses time to perform a task, especially a very

quick task (pdf spikes near 0 for small ) 2

Log-Normal

method of generation

let YeX

generate Y~N( , ) 2

Poisson counts the number of events that occur in a unit of

time when events are occurring at a constant rate

counts the number of events that occur in a unit of time when events are occuring with exponential inter-occurrence times

k

1ii 1Y:kmaxX

so, we can count events occurring before 1 unit of time by

where Yi are iid exponentials.

Poisson Specifically, to generate a Poisson rv with rate ,we

will generate exponential rate inter- arrival times.

Note that if U~unif(0,1),

)exp(rate ~ U)-ln(1 1

-Y

So, we also know that

)exp(rate ~ ln(U) 1

-Y

Poisson Algorithm:

Let a=e-1, b=1, counter=0

Generate U~unif(0,1) and let b=bU

• If b<a, done: return counter

• otherwise, counter = counter+1