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.

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