Binomial distribution
High school
Consider $n$ independent trials, each of which succeeds with probability $p$. The number $X$ of successes has a binomial distribution with parameters $n$ and $p$:
\[P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} \quad \text{for } k = 0, 1, \dots, n.\]Properties
- A binomial random variable is the sum of $n$ independent Bernoulli random variables, so it has mean $n p$ and variance $n p (1 - p)$.
- For large $n$, the binomial distribution is approximately normal, and for large $n$ and small $p$, the binomial distribution is approximately Poisson with mean $n p$.