Index: The Book of Statistical ProofsProbability Distributions ▷ Univariate discrete distributions ▷ Binomial distribution ▷ Mean

Theorem: Let $X$ be a random variable following a binomial distribution:

\[\label{eq:bin} X \sim \mathrm{Bin}(n,p) \; .\]

Then, the mean or expected value of $X$ is

\[\label{eq:bin-mean} \mathrm{E}(X) = n p \; .\]

Proof: By definition, a binomial random variable is the sum of $n$ independent and identical Bernoulli trials with success probability $p$. Therefore, the expected value is

\[\label{eq:bin-mean-s1} \mathrm{E}(X) = \mathrm{E}(X_1 + \ldots + X_n)\]

and because the expected value is a linear operator, this is equal to

\[\label{eq:bin-mean-s2} \begin{split} \mathrm{E}(X) &= \mathrm{E}(X_1) + \ldots + \mathrm{E}(X_n) \\ &= \sum_{i=1}^{n} \mathrm{E}(X_i) \; . \end{split}\]

With the expected value of the Bernoulli distribution, we have:

\[\label{eq:bin-mean-s3} \mathrm{E}(X) = \sum_{i=1}^{n} p = n p \; .\]
Sources:

Metadata: ID: P23 | shortcut: bin-mean | author: JoramSoch | date: 2020-01-16, 11:06.