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Is the probability density function (pdf) unique?

For example, I've seen the pdf of the uniform distribution written in two different versions, one with strict inequality and the other is not strict. From the definition, pdf is a function $f$ such that $P(X\in B)=\int_B f$. So it seems that the value of $x$ at one particular point doesn't really matter. Am I correct?

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Depends. If your pdf has a point mass somewhere, it would matter. For example, a mixture of half a uniform distribution, and a $0.5$ point mass at $x$, then the CDF would have a "jump" at $x$. – Memming Mar 18 '12 at 16:09

Indeed. If two (measurable) functions coincide except on a set of measure zero, both can be or cannot be equally chosen as density of the probability distribution of a given random variable.

For example, consider the Borel function $f$ defined on $\mathbb R$ by $f(x)=1$ for every irrational $x$ in $(0,1)$ and $f(x)=0$ otherwise. For every random variable $X$ uniformly distributed on $[0,1]$ and every Borel subset $B$ of $\mathbb R$, $\mathrm P(X\in B)=\int\limits_Bf(x)\mathrm dx$. Hence the function $f$ is indeed a density for the uniform distribution on $[0,1]$, as suitable as the commoner choices $\mathbf 1_{[0,1]}$ or $\mathbf 1_{(0,1)}$.

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