Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
    
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
add comment

1 Answer

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 $\mathbf 1_{(0,1)}$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.