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Glossary

Binomial random variable

A binomial random variable is a random variable that counts the results of n "binomial trials." A binomial trial is simply a much simpler random variable that takes the value 1 with probability p and takes the value 0 with probability 1p.

The simplest type of binomial trial is the flip of a fair coin (which corresponds to p=1/2). For this reason, other binomial trials may be thought of as weighted coin flips.

For n binomial trials with probability p, a binomial random variable Bin(n,p) counts the number of trials taking the value 1. One may verify that if X is such a random variable, then Pr(X=k) is given by \binom{n}{k} p^k \cdot (1-p)^{n-k}.

The chart below illustrates three different binomial random variables over 20 trials. In each case, k is plotted against \mathrm{Pr}(X = k) as k ranges between 0 and 20. The blue chart represents p = 0.1, the green chart represents p = 0.5, and the red chart represents p = 0.8.

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