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Probability distribution value exceeding 1 is OK?

I'm a bit confused how I am getting probabilities greater than 1 when calculating p(x | mu, sigma) when x = mu. For example, if I run:

>> gaussProb(0, 0, 0.1)
ans =

where gaussProb is a matlab function from the PMTK toolbox:

function p = gaussProb(X, mu, Sigma)
% Multivariate Gaussian distribution, pdf
% X(i,:) is i'th case
% *** In the univariate case, Sigma is the variance, not the standard
% deviation! ***

% This file is from pmtk3.googlecode.com

d = size(Sigma, 2);
X  = reshape(X, [], d);  % make sure X is n-by-d and not d-by-n
X = bsxfun(@minus, X, rowvec(mu));
logp = -0.5*sum((X/(Sigma)).*X, 2); 
logZ = (d/2)*log(2*pi) + 0.5*logdet(Sigma);
logp = logp - logZ;
p = exp(logp);        


Is this some fundamental property of the Gaussian distribution or an issue with numerical accuracy in the computation?

I've come across this issue by trying to weight samples from a Gaussian distribution obtained from a Gaussian process prediction, where I will get massive probabilities.



1 Answer 1


The code in the question returns the values of probability density function. The values of probability density function can be greater than one. The actual probability $P(X<x)$ for random variable $X$ with probability density function $p(x)$ is integral $\int_{-\infty}^xp(t)dt$. The values of this integral are of course restricted to interval $[0,1]$.


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