# Why does it make sense to define the Probability mass function of e.g. the binomial distribution like this?

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

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

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

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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Could you be more specific about where you saw this notation? Also, in your second equation, the right hand side is a constant independent of $k$ since $k$ is a dummy variable. Can you double check that equation? – Srivatsan Jan 23 '12 at 8:50
@Srivatsan: There was a typo, sorry. It is script of a german professor. I'll give the link anyway. You can find the notation in this pdf on page 44. – Aufwind Jan 23 '12 at 8:53
After the recent edits: Now I see the intended meaning. $\delta_k$ is not a variable as I assumed, but a function that is $1$ at $k$ and $0$ everywhere else. This is mostly for notational convenience; also manipulating algebraic expressions becomes easier using the second notation as compared to the first. – Srivatsan Jan 23 '12 at 9:00
Rather than a function, each $\delta_k$ is a (probability) measure (as $P_{n,p}$). – Did Jan 23 '12 at 9:56

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 $A \subseteq \mathbb{N}$, $$P_{n,p}(A) = \sum_{k=0}^n\binom{n}{k}p^k(1-p)^{n-k} \delta_k(A) = \sum_{k \in \{0,1,\cdots,n\}\cap A} \binom{n}{k}p^k(1-p)^{n-k}$$

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