Proof That the Binomial Distribution is a Probability Distribution
In order to be a probability distribution for a random variable, a function must sum or integrate over all the possible values of the random variable to 1.If a random variable
\[X\]
is modelled by the binomial distribution, \[X \sim B(n,p)\]
then \[P(X=x) = \frac{n!}{x! (n-x)!} p^x (1-p)^{n-x}\]
The binomial expansion tell us
\[\begin{equation} \begin{aligned} (a+b)^n &= a^n + n a^{n-1} b + \frac{n(n-1)}{2} a^{n-2} b^2 +...+ \frac{(n-1)n}{2}a^2 b^{n-2} + (n-1) ab^{n-1} + b^n \\ &= \sum_{k=0}^n \frac{n!}{k!(n-k)!} a^k b^{n-k} \end{aligned} \end{equation}\]
Substitute
\[a=p, b=1-p\]
then\[ \sum_{k=0}^n \frac{n!}{k!(n-k)!} p^k (1-p)^{n-k} =(p+1-p)^n =1 \]