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A. The Binomial Distribution

This you have seen in CS1021. It is also called a Bernoulli distribution. It is applicable to the situation where there are $N$ independent trials, each of which can have one of two outcomes. Call one of the outcomes $0$ and the other $1$. Let $p$ be the probability that any trial yields the outcome $1$. Then the probability of getting $n$ 1's in $N$ trials is given by

\begin{displaymath}P(n\vert p,N) = {N \choose n} p^n (1-p)^{N-n} ,\end{displaymath}


\begin{displaymath}{N \choose n} \equiv \frac{N!}{n!(N-n)!} .\end{displaymath}

The expected mean number of 1's is $N p$; the variance is $N p (1-p)$

In the limit of large $N$ with $p$ fixed, a binomial distribution is well approximated by a continuous Gaussian distribution, with the appropriate mean and variance.

Jon Shapiro