# Does the difference of the total happened count of two incidents of a bernoulli distributions also follows certain distribution?

Oh, it is hard to express math questions in English. What I mean concerns a Bernoulli distribution $B(n,p)$. We know that $A$ happens $k$ times and $A^c$ happens $n-k$ times. So does the difference $(k-(n-k))=2k-n$ also follow certain distribution? And what is the distribution?

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This is sometimes called "the heads minus tails" distribution, although that name is often also used for the distribution of the absolute value of the difference. It is very closely related to the binomial distribution. The probabilities can be simply written down from the known probabilities for the binomial. – André Nicolas Jan 17 '13 at 6:09

This random variable can also be described as the sum of $n$ independent random variables each of which takes one of values $\pm 1$ with probabilities $p$ and $1-p$. When $n$ is large, the resulting distribution is approximately Gaussian, $\mathcal{N}((1-2p)n,2(1-p)\sqrt{n})$. The exact distribution is $$P(X=m)=\begin{cases} p^{(n+m)/2}(1-p)^{(n-m)/2}\binom{n}{(n+m)/2} \quad &\text{if }\ (n+m)/2 \in \{0,1,\dots,n\} \\ 0 &\text{otherwise}\end{cases}$$ For example, with $n=40$ and $p=1/2$ it already looks pretty much normal: