---
title: "Binomial distribution"
sort_title: "Binomial distribution"
date: 2021-02-26
categories:
- Statistics
- Mathematics
layout: "concept"
---

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:

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


{% include proof/start.html id="proof-mean" -%}
The trick is to treat $$p$$ and $$q$$ as independent and introduce a derivative:

$$\begin{aligned}
    \mu
    &= \sum_{n = 0}^N n P_N(n)
    = \sum_{n = 0}^N n \binom{N}{n} p^n q^{N - n}
    = \sum_{n = 0}^N \binom{N}{n} \bigg( p \pdv{(p^n)}{p} \bigg) q^{N - n}
\end{aligned}$$

Then, using the fact that the binomial coefficients appear when writing out $$(p + q)^N$$:

$$\begin{aligned}
    \mu
    &= p \pdv{}{p}\sum_{n = 0}^N \binom{N}{n} p^n q^{N - n}
    = p \pdv{}{p}(p + q)^N
    = N p (p + q)^{N - 1}
\end{aligned}$$

Finally, inserting $$q = 1 - p$$ gives the desired result.
{% include proof/end.html id="proof-mean" %}


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

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


{% include proof/start.html id="proof-var" -%}
We reuse the previous trick to find $$\overline{n^2}$$
(the mean squared number of successes):

$$\begin{aligned}
    \overline{n^2}
    &= \sum_{n = 0}^N n^2 \binom{N}{n} p^n q^{N - n}
    = \sum_{n = 0}^N n \binom{N}{n} \bigg( p \pdv{}{p} \bigg) p^n q^{N - n}
    \\
    &= \sum_{n = 0}^N \binom{N}{n} \bigg( p \pdv{}{p} \bigg)^2 p^n q^{N - n}
    = \bigg( p \pdv{}{p} \bigg)^2 \sum_{n = 0}^N \binom{N}{n} p^n q^{N - n}
    \\
    &= \bigg( p \pdv{}{p} \bigg)^2 (p + q)^N
    = N p \pdv{}{p}p (p + q)^{N - 1}
    \\
    &= N p \big( (p + q)^{N - 1} + (N - 1) p (p + q)^{N - 2} \big)
    \\
    &= N p + N^2 p^2 - N p^2
\end{aligned}$$

Using this and the earlier expression $$\mu = N p$$, we find the variance $$\sigma^2$$:

$$\begin{aligned}
    \sigma^2
    &= \overline{n^2} - \mu^2
    = N p + N^2 p^2 - N p^2 - N^2 p^2
    = N p (1 - p)
\end{aligned}$$

By inserting $$q = 1 - p$$, we arrive at the desired expression.
{% include proof/end.html id="proof-var" %}


As $$N \to \infty$$, the binomial distribution
turns into the continuous normal distribution,
a fact that is sometimes called the **de Moivre-Laplace theorem**:

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


{% include proof/start.html id="proof-normal" -%}
We take the Taylor expansion of $$\ln\!\big(P_N(n)\big)$$
around the mean $$\mu = Np$$:

$$\begin{aligned}
    \ln\!\big(P_N(n)\big)
    &= \sum_{m = 0}^\infty \frac{(n - \mu)^m}{m!} D_m(\mu)
    \quad \mathrm{where} \quad
    D_m(n)
    \equiv \dvn{m}{\ln\!\big(P_N(n)\big)}{n}
\end{aligned}$$

For future convenience while calculating the $$D_m$$, we write out $$\ln(P_N)$$ now:

$$\begin{aligned}
    \ln\!\big(P_N(n)\big)
    &= \ln(N!) - \ln(n!) - \ln\!\big((N \!-\! n)!\big) + n \ln(p) + (N \!-\! n) \ln(q)
\end{aligned}$$

For $$D_0(\mu)$$ specifically,
we need to use a strong version of *Stirling's approximation*
to arrive at a nonzero result in the end.
We know that $$N - N p = N q$$:

$$\begin{aligned}
    D_0(\mu)
    &= \ln\!\big(P_N(n)\big) \big|_{n = \mu}
    \\
    &= \ln(N!) - \ln(\mu!) - \ln\!\big((N \!-\! \mu)!\big) + \mu \ln(p) + (N \!-\! \mu) \ln(q)
    \\
    &= \ln(N!) - \ln\!\big((N p)!\big) - \ln\!\big((N q)!\big) + N p \ln(p) + N q \ln(q)
    \\
    &\approx \Big( N \ln(N) - N + \frac{1}{2} \ln(2\pi N) \Big)
    - \Big( N p \ln(N p) - N p + \frac{1}{2} \ln(2\pi N p) \Big) \\
    &\qquad - \Big( N q \ln(N q) - N q + \frac{1}{2} \ln(2\pi N q) \Big)
    + N p \ln(p) + N q \ln(q)
    \\
    &= N \ln(N) - N (p \!+\! q) \ln(N) + N (p \!+\! q) - N - \frac{1}{2} \ln(2\pi N p q)
    \\
    &= - \frac{1}{2} \ln(2\pi N p q)
    = \ln\!\bigg( \frac{1}{\sqrt{2\pi \sigma^2}} \bigg)
\end{aligned}$$

Next, for $$D_m(\mu)$$ with $$m \ge 1$$,
we can use a weaker version of Stirling's approximation:

$$\begin{aligned}
    \ln(P_N)
    &\approx \ln(N!) - n \big( \ln(n) \!-\! 1 \big) - (N \!-\! n) \big( \ln(N \!-\! n) \!-\! 1 \big) + n \ln(p) + (N \!-\! n) \ln(q)
    \\
    &\approx \ln(N!) - n \big( \ln(n) - \ln(p) - 1 \big) - (N\!-\!n) \big( \ln(N\!-\!n) - \ln(q) - 1 \big)
\end{aligned}$$

We expect that $$D_1(\mu) = 0$$, because $$P_N$$ is maximized at $$\mu$$.
Indeed it is:

$$\begin{aligned}
    D_1(n)
    &= \dv{}{n} \ln\!\big((P_N(n)\big)
    \\
    &= - \big( \ln(n) - \ln(p) - 1 \big) + \big( \ln(N\!-\!n) - \ln(q) - 1 \big) - \frac{n}{n} + \frac{N \!-\! n}{N \!-\! n}
    \\
    &= - \ln(n) + \ln(N \!-\! n) + \ln(p) - \ln(q)
    \\
    D_1(\mu)
    &= - \ln(\mu) + \ln(N \!-\! \mu) + \ln(p) - \ln(q)
    \\
    &= - \ln(N p q) + \ln(N p q)
    \\
    &= 0
\end{aligned}$$

For the same reason, we expect $$D_2(\mu)$$ to be negative.
We find the following expression:

$$\begin{aligned}
    D_2(n)
    &= \dvn{2}{}{n} \ln\!\big((P_N(n)\big)
    = \dv{}{n} D_1(n)
    = - \frac{1}{n} - \frac{1}{N - n}
    \\
    D_2(\mu)
    &= - \frac{1}{Np} - \frac{1}{Nq}
    = - \frac{p + q}{N p q}
    = - \frac{1}{\sigma^2}
\end{aligned}$$

The higher-order derivatives vanish much faster as $$N \to \infty$$, so we discard them:

$$\begin{aligned}
    D_3(n)
    = \frac{1}{n^2} - \frac{1}{(N - n)^2}
    \qquad \quad
    D_4(n)
    = - \frac{2}{n^3} - \frac{2}{(N - n)^3}
    \qquad \quad
    \cdots
\end{aligned}$$

Putting everything together, for large $$N$$,
the Taylor series approximately becomes:

$$\begin{aligned}
    \ln\!\big(P_N(n)\big)
    \approx D_0(\mu) + \frac{(n - \mu)^2}{2} D_2(\mu)
    = \ln\!\bigg( \frac{1}{\sqrt{2\pi \sigma^2}} \bigg) - \frac{(n - \mu)^2}{2 \sigma^2}
\end{aligned}$$

Raising $$e$$ to this expression then yields a normalized Gaussian distribution.
{% include proof/end.html id="proof-normal" %}



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