binomial distribution

A binomial random variable is a discrete random variable counting the number of "success" outcomes in a binomial experiment. Such a random variable follows a binomial distribution, denoted $X \sim Binom (n, p)$ , where $n \in N$ is the number of trials and $p \in [0, 1]$ is the probability of success in each one.

Formally,
if $X$ is a binomial random variable, i.e. $X \sim Binom (n, p)$ , then the probability of $k$ successes is given by the probability mass function

b (k; n, p) = P (X = k) = (\binom{n}{k}) p^{k} (1 - p)^{n - k}

and its cumulative distribution function is

B (k; n, p) = P (X \leq k) = \sum_{i = 0}^{k} (\binom{n}{i}) p^{i} (1 - p)^{n - i}

where $(\binom{n}{k})$ is the binomial coefficient, defined such that $(\binom{n}{k}) = 0$ for $k \notin [0, n]$ .

If $X \sim Binom (n, p)$ , then

the expected value $E [X] = n p$
the variance $Var (X) = n p (1 - p)$