Proof: Mean of the binomial distribution

Index: The Book of Statistical Proofs ▷ Probability 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

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

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

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

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Sources:

• Wikipedia (2020): "Binomial distribution"; in: Wikipedia, the free encyclopedia, retrieved on 2020-01-16; URL: https://en.wikipedia.org/wiki/Binomial_distribution#Expected_value_and_variance.

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