LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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A Bernoulli $(p)$ variable has mean $p$ and variance $p (1 - p)$ ; a binomial $(n, p)$ variable has mean $n p$ and variance $n p (1 - p)$

Statement

If $X$ is Bernoulli $(p)$ , then $E [X] = p$ and $Var (X) = p (1 - p)$ . If $S$ is binomial $(n, p)$ , then $E [S] = n p, Var (S) = n p (1 - p) .$ These formulas include $p = 0$ , $p = 1$ , and $n = 0$ .

Facts & Assumptions

Given: A Bernoulli $(p)$ variable $X$ and a binomial $(n, p)$ variable $S = \sum_{i < n} X_{i}$ .

[L1]

Bernoulli and binomial variables have the distributions and independent-sum form in the Definition (Bernoulli random variables and binomial random variables as sums of independent Bernoulli trials).

[L2]

Expectation is linear without independence (Expectation is linear for every finite family of random variables, without any independence hypothesis).

[L3]

$Var (X) = E [X^{2}] - E [X]^{2}$ ( $Var (X) = E [X^{2}] - E [X]^{2}$ and $Cov (X, Y) = E [X Y] - E [X] E [Y]$ ).

[L4]

Variance adds for finite pairwise-independent families (Variance adds for every finite pairwise-independent family).

Proof

technique · direct

1.1

Since $X$ is $1$ with probability $p$ and $0$ otherwise, $E [X] = p$ and $E [X^{2}] = p$ .

2.1

Hence $Var (X) = p - p^{2} = p (1 - p)$ .

step 1.1L3algebra

3.1

Linearity gives $E [S] = \sum_{i < n} p = n p$ , and pairwise-independent variance additivity gives $Var (S) = \sum_{i < n} p (1 - p) = n p (1 - p)$ .

step 1.1step 2.1L1L2L4algebra

4.1

At $p = 0$ or $p = 1$ the variance formula is zero, and at $n = 0$ both finite sums are empty and equal zero.

step 3.1∎

Depends on

Used by

First- and second-moment bounds for a nonempty Bernoulli random subset Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 39 results over 18 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Section 6.2 (standard reference, not scraped)
H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Sections 3.1.5 and 3.2.4 (standard reference, not scraped)

A Bernoulli(p)(p) variable has mean pp and variance p(1−p)p(1−p); a binomial(n,p)(n,p) variable has mean npnp and variance np(1−p)np(1−p)