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Methods for generating pseudorandom numbers usually start with
uniform random numbers, like the MATLAB^{®} `rand`

function
produces. The methods described in this section detail how to produce
random numbers from other distributions.

Direct methods directly use the definition of the distribution.

For example, consider binomial random numbers. A binomial
random number is the number of heads in *N* tosses
of a coin with probability *p* of a heads on any
single toss. If you generate *N* uniform random numbers
on the interval (0,1) and count the number less than *p*,
then the count is a binomial random number with parameters *N* and *p*.

This function is a simple implementation of a binomial RNG using the direct approach:

function X = directbinornd(N,p,m,n) X = zeros(m,n); % Preallocate memory for i = 1:m*n u = rand(N,1); X(i) = sum(u < p); end end

For example:

X = directbinornd(100,0.3,1e4,1); histogram(X,101)

The Statistics and Machine
Learning Toolbox™ function `binornd`

uses
a modified direct method, based on the definition of a binomial random
variable as the sum of Bernoulli random variables.

You can easily convert the previous method to a random number
generator for the Poisson
distribution with parameter *λ*. The
Poisson distribution is the limiting case of the binomial distribution
as *N* approaches infinity, *p* approaches
zero, and *Np* is held fixed at *λ*.
To generate Poisson random numbers, create a version of the previous
generator that inputs *λ* rather than *N* and *p*,
and internally sets *N* to some large number and *p* to *λ*/*N*.

The Statistics and Machine
Learning Toolbox function `poissrnd`

actually
uses two direct methods:

A waiting time method for small values of

*λ*A method due to Ahrens and Dieter for larger values of

*λ*

*Inversion methods* are based on the observation
that continuous cumulative distribution functions (cdfs) range uniformly
over the interval (0,1). If *u* is a uniform random
number on (0,1), then using *X* = *F* ^{-1}(*U*) generates
a random number *X* from a continuous distribution
with specified cdf *F*.

For example, the following code generates random numbers from
a specific exponential
distribution using the inverse cdf and the MATLAB uniform
random number generator `rand`

:

mu = 1; X = expinv(rand(1e4,1),mu);

Compare the distribution of the generated random numbers to the pdf of the specified exponential by scaling the pdf to the area of the histogram used to display the distribution:

numbins = 50; h = histogram(X,numbins) hold on histarea = h.BinWidth*sum(h.Values); x = h.BinEdges(1):0.001:h.BinEdges(end); y = exppdf(x,mu); plot(x,histarea*y,'r','LineWidth',2)

Inversion methods also work for discrete distributions. To generate
a random number *X* from a discrete distribution
with probability mass vector *P*(*X*=*x*_{i}) = *p*_{i} where *x*_{0}<*x*_{1}<* x*_{2}<..., generate a uniform random number *u* on
(0,1) and then set *X* = *x*_{i} if * F*(*x*_{i–1})<*u*<*F*(*x*_{i}).

For example, the following function implements an inversion
method for a discrete distribution with probability mass vector *p*:

function X = discreteinvrnd(p,m,n) X = zeros(m,n); % Preallocate memory for i = 1:m*n u = rand; I = find(u < cumsum(p)); X(i) = min(I); end end

Use the function to generate random numbers from any discrete distribution:

```
p = [0.1 0.2 0.3 0.2 0.1 0.1]; % Probability mass vector
X = discreteinvrnd(p,1e4,1);
h = histogram(X,length(p));
bar(1:length(p),h.Values)
```

The functional form of some distributions makes it difficult or time-consuming to generate random numbers using direct or inversion methods. Acceptance-rejection methods provide an alternative in these cases.

*Acceptance-rejection methods* begin with
uniform random numbers, but require an additional random number generator.
If your goal is to generate a random number from a continuous distribution
with pdf *f*, acceptance-rejection methods first
generate a random number from a continuous distribution with pdf *g* satisfying *f*(*x*)
≤ *c**g*(*x*) for some *c* and all *x*.

A continuous acceptance-rejection RNG proceeds as follows:

Chooses a density

*g*.Finds a constant

*c*such that*f*(*x*)/*g*(*x*)≤*c*for all*x*.Generates a uniform random number

*u*.Generates a random number

*v*from*g*.If

*c**u*≤*f*(*v*)/*g*(*v*), accepts and returns*v*.Otherwise, rejects

*v*and goes to step 3.

For efficiency, a “cheap” method is necessary
for generating random numbers from *g*, and the
scalar *c* should be small. The expected number
of iterations to produce a single random number is *c*.

The following function implements an acceptance-rejection method
for generating random numbers from pdf *f*, given *f*, *g*,
the RNG `grnd`

for *g*, and the
constant *c*:

function X = accrejrnd(f,g,grnd,c,m,n) X = zeros(m,n); % Preallocate memory for i = 1:m*n accept = false; while accept == false u = rand(); v = grnd(); if c*u <= f(v)/g(v) X(i) = v; accept = true; end end end

For example, the function *f*(*x*)
= *x*e^{–x2/2} satisfies
the conditions for a pdf on [0,∞) (nonnegative and integrates
to 1). The exponential pdf with mean 1, *f*(*x*)
= e^{–x},
dominates *g* for *c* greater
than about 2.2. Thus, you can use `rand`

and `exprnd`

to generate random numbers from *f*:

f = @(x)x.*exp(-(x.^2)/2); g = @(x)exp(-x); grnd = @()exprnd(1); X = accrejrnd(f,g,grnd,2.2,1e4,1);

The pdf *f* is actually a Rayleigh distribution with
shape parameter 1. This example compares the distribution of random
numbers generated by the acceptance-rejection method with those generated
by `raylrnd`

:

Y = raylrnd(1,1e4,1); histogram(X) hold on histogram(Y) legend('A-R RNG','Rayleigh RNG')

The Statistics and Machine
Learning Toolbox function `raylrnd`

uses
a transformation method, expressing a Rayleigh random variable in
terms of a chi-square random
variable, which you compute using `randn`

.

Acceptance-rejection methods also work for discrete distributions.
In this case, the goal is to generate random numbers from a distribution
with probability mass *P*_{p}(*X* = *i*) = *p*_{i}, assuming that you have a method for generating random
numbers from a distribution with probability mass *P*_{q}(*X* = *i*) = *q*_{i}. The RNG proceeds as follows:

Chooses a density

.*P*_{q}Finds a constant

*c*such that*p*_{i}/*q*_{i}≤*c*for all*i*.Generates a uniform random number

*u*.Generates a random number

*v*from.*P*_{q}If

*c**u*≤*p*_{v}/*q*_{v}, accepts and returns*v*.Otherwise, rejects

*v*and goes to step 3.

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