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In Wikipedia (for example) the Probability mass function of for example a the binomial distribution is given by

$ f(k,n,p):=\binom{n}{k}p^k(1-p)^{n-k} $

In some literature I read

$ P_{n,p}:=\sum_{k=0}^n\binom{n}{k}p^k(1-p)^{n-k} \delta_k $

where $\delta_k$ is the Dirac delta function.

I see that the result is the same, because $\delta_k$ nullifies each not needed addend. Why this verbosity? Does it have a sense, that I didn't get so far?

Thanks for any feedback!

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    Rather than a function, each $\delta_k$ is a (probability) measure (as $P_{n,p}$).2012-01-23

3 Answers 3

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The Wikipedia page considers a discrete probability distribution defined on the natural numbers. The use of the Dirac measure $\delta_k$ suggests that a probability density function of a continuous random variable is defined. In the latter setting the former definition would not do, because $\tbinom nk$ can be given a non-zero meaning at non-integral $k$ using the Beta function (or using the Gamma function if you prefer), and this would not be the right thing for getting a probability distribution.

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As Didier Piau hinted, the first expression is a density function for your random variable (implicitely distributed on the integers), while the second expression is the law of your random variable.

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Here $\delta_k$ is a set-function defined such that for any $A \subseteq \mathbb{N}$, $\delta_k(A) = \left\{ \begin{gathered} {1} \quad{\text{if}}\quad k \in A \\ {0} \quad{\text{if}}\quad k \notin A \\ \end{gathered} \right.$ i.e., expressed as an Iverson bracket, $\delta_k(A) = [k \in A]$.

Thus, the probability measure $P_{n,p}$ is such that for any event $A \subseteq \mathbb{N}$, $ P_{n,p}(A) = \sum_{k=0}^n\binom{n}{k}p^k(1-p)^{n-k} \delta_k(A) = \sum_{k \in \{0,1,\cdots,n\}\cap A} \binom{n}{k}p^k(1-p)^{n-k} $