Editing Binomial distribution (section)

=== Expected value and variance ===

If {{math|''X'' ~ ''B''(''n'', ''p'')}}, that is, {{math|''X''}} is a binomially distributed random variable, {{mvar|n}} being the total number of experiments and ''p'' the probability of each experiment yielding a successful result, then the [[expected value]] of {{math|''X''}} is:<ref>See [https://proofwiki.org/wiki/Expectation_of_Binomial_Distribution Proof Wiki]</ref>
: <math> \operatorname{E}[X] = np.</math>

This follows from the linearity of the expected value along with the fact that {{mvar|X}} is the sum of {{mvar|n}} identical Bernoulli random variables, each with expected value {{mvar|p}}.  In other words, if <math>X_1, \ldots, X_n</math> are identical (and independent) Bernoulli random variables with parameter {{mvar|p}}, then {{math|1=''X'' = ''X''<sub>1</sub> + ... + ''X''<sub>''n''</sub>}} and
: <math>\operatorname{E}[X] = \operatorname{E}[X_1 + \cdots + X_n] = \operatorname{E}[X_1] + \cdots + \operatorname{E}[X_n] = p + \cdots + p = np.</math>

The [[variance]] is:
: <math> \operatorname{Var}(X) = npq = np(1 - p).</math>

This similarly follows from the fact that the variance of a sum of independent random variables is the sum of the variances.