Expectation: Binomial (Bernoulli) distribution

September, 2026

For a random variable $X_{i}$ that determines the outcome of an experiment, for example, bit array generation with the probabiliy for $1$ being $p$ and $0$ being $1 - p$ we observe the following:

P r [X_{i} = 1] P r [X_{i} = 0] = p = 1 - p

By definition, the expected value for $X_{i}$ is

E x [X_{i}] = 1 \cdot p + 0 \cdot (1 - p) = p

for positive outcomes where $X_{i} = 1$ with the probability $p$ .

Random variable $X$ gives us the total amount of positive outcomes during the experiment and thus it's value is:

X = X_{1} + X_{2} + \dots + X_{n}

Now, given the linearity of expectation we get:

E x [X] = E x [X_{1} + X_{2} + \dots + X_{n}] = E x [X_{1}] + E x [X_{2}] + \dots + E x [X_{n}]

Finally, we know that the $E x [X_{i}] = p$ for each $i$ and

E x [X] = p + p + \dots + p = n \cdot p

Which gives us an easy way to evaluate an expectation for random variable $X$ that has a Binomial distribution, thus for $X : B (n, p)$ :

E x [X] = n \cdot p

Formulas

E x [X] = n \cdot p

Va r [X] = n \cdot p \cdot (1 - p)

Appendix

-Distributions: Bernoulli