Categories: Mathematics, Statistics.

# Binomial distribution

The binomial distribution is a discrete probability distribution describing a Bernoulli process: a set of independent $$N$$ trials where each has only two possible outcomes, “success” and “failure”, the former with probability $$p$$ and the latter with $$q = 1 - p$$. The binomial distribution then gives the probability that $$n$$ out of the $$N$$ trials succeed:

\begin{aligned} \boxed{ P_N(n) = \binom{N}{n} \: p^n q^{N - n} } \end{aligned}

The first factor is known as the binomial coefficient, which describes the number of microstates (i.e. permutations) that have $$n$$ successes out of $$N$$ trials. These happen to be the coefficients in the polynomial $$(a + b)^N$$, and can be read off of Pascal’s triangle. It is defined as follows:

\begin{aligned} \boxed{ \binom{N}{n} = \frac{N!}{n! (N - n)!} } \end{aligned}

The remaining factor $$p^n (1 - p)^{N - n}$$ is then just the probability of attaining each microstate.

The expected or mean number of successes $$\mu$$ after $$N$$ trials is as follows:

\begin{aligned} \boxed{ \mu = N p } \end{aligned}

Meanwhile, we find the following variance $$\sigma^2$$, with $$\sigma$$ being the standard deviation:

\begin{aligned} \boxed{ \sigma^2 = N p q } \end{aligned}

As $$N \to \infty$$, the binomial distribution turns into the continuous normal distribution:

\begin{aligned} \boxed{ \lim_{N \to \infty} P_N(n) = \frac{1}{\sqrt{2 \pi \sigma^2}} \exp\!\Big(\!-\!\frac{(n - \mu)^2}{2 \sigma^2} \Big) } \end{aligned}

## References

1. H. Gould, J. Tobochnik, Statistical and thermal physics, 2nd edition, Princeton.

© Marcus R.A. Newman, a.k.a. "Prefetch". Available under CC BY-SA 4.0.