Binomial distribution

The binomial distribution is a probability mass function that generalises the Bernoulli distribution for $n$ trials if:

• The trials are independent.
• Each trial has the same success probability $p$.
• $X$ is the number of successes in the $n$ trials.

We denote it with:

$$X\sim B(n,p)$$

where $X\in [0,n]$, a discrete random variable.

Below are the distributions for $B(10,0.4)$ and $B(20,0.1)$.¹

Calculations

The probability mass function of $X$ is given by:

$$p(x)=P(X=x)=\{\begin{array}{ll}\frac{n!}{x!(n-x)!}{p}^x(1-p{)}^{n-x}& x=0,1,\dots ,n\\ 0& \text{otherwise}\end{array}$$

The mean and variance are given by:

$${\mu }_X=np$$ $${\sigma }_X^2=np(1-p)$$

Footnotes

1. From Statistics for Engineers and Scientists, by William Navidi. ↩