MATLAB Answers

Poisson random number generator

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Ahmed raheem
Ahmed raheem on 6 Feb 2012
Commented: Ian Van Giesen on 24 Jun 2020 at 21:15
Hi all please i need to know how to generate a Poisson distributed random variable without using the built-in function (poissrnd).

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Accepted Answer

Andreas Goser
Andreas Goser on 6 Feb 2012
If this is an acadamic exercise - you can look at the literature refererence
% References:
% [1] Devroye, L. (1986) Non-Uniform Random Variate Generation,
% Springer-Verlag.

More Answers (8)

Derek O'Connor
Derek O'Connor on 7 Feb 2012
I prefer this:
% -------------------------------------------------------------
function S = PoissonSamp(lambda,ns);
% -------------------------------------------------------------
% Generate a random sample S of size ns from the (discrete)
% Poisson distribution with parameter lambda.
% Derek O'Connor, 6 Feb 2012. derekroconnor@eircom.net
%
S = zeros(ns,1);
for i = 1:ns
k=1; produ = 1;
produ = produ*rand;
while produ >= exp(-lambda)
produ = produ*rand;
k = k+1;
end
S(i) = k;
end

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Derek O'Connor
Derek O'Connor on 3 May 2012
I would like to thank Kang Wook Lee of Berkeley for pointing out an error in the code above. The last line should be S(i) = k-1;
% -------------------------------------------------------------
function S = PoissonSamp(lambda,ns);
% -------------------------------------------------------------
% Generate a random sample S of size ns from the (discrete)
% Poisson distribution with parameter lambda.
% Fixed error: changed S(i) = k; to S(i) = k-1;
% Derek O'Connor, 3 May 2012. derekroconnor@eircom.net
%
S = zeros(ns,1);
for i = 1:ns
k=1; produ = 1;
produ = produ*rand;
while produ >= exp(-lambda)
produ = produ*rand;
k = k+1;
end
S(i) = k-1;
end

  2 Comments

Ian Van Giesen
Ian Van Giesen on 24 Jun 2020 at 21:15
Why was the last line changed? Was it to make the events where probability for 'success' a null event? Sorry in advance for a confusing question, still wrapping my head around this, but many thanks for the code!

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Derek O'Connor
Derek O'Connor on 24 Jul 2012
This is a cleaner fix of PoissonSamp
% -------------------------------------------------------------
function S = PoissonSamp3(lambda,ns);
% -------------------------------------------------------------
% Generate a random sample S of size ns from the (discrete)
% Poisson distribution with parameter lambda.
% Fixed error:
% CHANGED k = 1; produ = 1; produ = produ*rand
% TO k = 0; produ = rand;
% Derek O'Connor, 24 July 2012. derekroconnor@eircom.net
%
S = zeros(ns,1);
for i = 1:ns
k = 0;
produ = rand;
while produ >= exp(-lambda)
produ = produ*rand;
k = k+1;
end
S(i) = k;
end

  2 Comments

Robert
Robert on 14 Aug 2015
Thank you for providing this, this is extremely helpful! Does anyone know why the poisrnd function is no longer in MatLab?
John D'Errico
John D'Errico on 10 Jul 2019
Because the function name is spelled poissrnd, not poisrnd.

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Richard Willey
Richard Willey on 6 Feb 2012
Mark Steyvers has written a nice book titled "Computational Statistics with MATLAB" which can be downloaded from
The first chapter has a very good section describing inverse transform sampling which provides everything you need to know.

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Derek O'Connor
Derek O'Connor on 6 Feb 2012
Dirk Kroese has excellent notes here: http://www.maths.uq.edu.au/~kroese/mccourse.pdf, which are based on his book:
D.P. Kroese, T. Taimre, Z.I. Botev: Handbook of Monte Carlo Methods. John Wiley & Sons, 2011.
His notes and book have lots of Matlab examples.

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Ahmed raheem
Ahmed raheem on 7 Feb 2012
thank you for your help....
this one is quite helpful...

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Ahmed raheem
Ahmed raheem on 7 Feb 2012
First of all, I'd like to thank you all for your cooperation with me. Based on Dirk Kroese, I wrote the following code to generate the Poisson random variable:
The m-file code:
n=1;
lambda=500;
for i=1:10000
x=rand(1);
a=1;
a=a*x;
if a>=exp(-lambda)
n=n+1;
continue
else
X(i)=n-1;
end
end
  • would you please check it if it's correct or not? i feel it is not correct.

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Derek O'Connor
Derek O'Connor on 7 Feb 2012
@Ahmed, you're correct, it is not correct.
The function below is a Matlab translation of Kroese's algorithm. It seems to work ok but needs to be thoroughly tested. You should do this and let us know the results. Note that Poisson(L) ~ Norm(L,L), for large L.
The one thing I don't like about Kroese is his awful algorithm and programming style. Why does he use GOTOs instead of proper WHILEs etc?
% -------------------------------------------------------------
function X = Poisson(lambda);
% -------------------------------------------------------------
% Generate a random value from the (discrete) Poisson
% distribution with parameter lambda.
% Derek O'Connor, 6 Feb 2012. derekroconnor@eircom.net
%
k=1; produ = 1;
produ = produ*rand;
while produ >= exp(-lambda)
produ = produ*rand;
k = k+1;
end
X = k;

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Ahmed raheem
Ahmed raheem on 7 Feb 2012
thank you Derek.
it is performing well now...
i putted the code inside a (for loop) and the result was ok.
thanks again.

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Ahmed raheem
Ahmed raheem on 7 Feb 2012
This is the final code:
% -------------------------------------------------------------
function X = Poisson(lambda,n); % n represents the number of iterations
% -------------------------------------------------------------
% Generate a random value from the (discrete) Poisson
% distribution with parameter lambda.
% Derek O'Connor, 6 Feb 2012. derekroconnor@eircom.net
%
for i=1:n;
k=1; usave=1;
usave = usave*rand;
while usave >= exp(-lambda)
usave = usave*rand;
k = k+1;
end
X(i) = k;
end
hist(X)

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