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Most of the books and sites define Probability function for discrete case that is they use the term as the synonym of Probability mass function.

Is that Probability function define for only discrete case?

Is Probability density function not included in Probability function?

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up vote 2 down vote accepted

A probability mass function (PMF) is used to describe a probability measure which is completely discrete, that is there exist countably many $\omega \in \Omega$ such that $P[\{\omega\}]>0$, and any subset which does not include one of these $\omega$ has probability 0.

A probability density function (PDF) is used to describe a probability measure (on $\mathbb{R}$, typically of the form $P\circ X^{-1}$ for a random variable $X$) which is absolutely continuous. In this case we have $P[A] = \int_A f(x)dx$ and call $f$ the PDF.

Note that these two cases are disjoint, if there is a PMF there cannot be a PDF and if there is a PDF then there cannot be a PMF. To see this, note that if a PMF exists then there is a singleton with strictly positive probability, and if a PDF exists then every singleton has zero probability.

The underlying theme here is that we are trying to describe the probability measure $P$. Since $P$ can be very complicated in general, we make special distinctions for the case of when a PMF or PDF for $P$ exists, as these situations are much easier to deal with.

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in examination, i was asked to define probability function. There it was not distinct whether i would define PMF or PDF. In such case, what should i define? – time Jun 17 '13 at 15:21
The term "probability function" is ambiguous and should never be used without context. It could refer to a CDF, PDF, PMF, or even the probability measure $P$ itself. Unless you are following a textbook which specifically defines the term, I would avoid it. As far as your examination goes, I would point this out to your instructor. – nullUser Jun 17 '13 at 15:24
My only guess would be that if you are taking a course which is exclusively on discrete probability, then "probability function" most likely means PMF. – nullUser Jun 17 '13 at 15:25
Thank you very much. Well explained. – time Jun 17 '13 at 15:27

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