4.11 Parameters for a Binomial Distribution

Part of Probability, Random Variables, and Probability Distributions in AP® Statistics. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

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Binomial Mean and Standard Deviation

For ${X \sim B(n, p)}$ :

\mu_X = np

\sigma_X^2 = np(1-p)

\sigma_X = \sqrt{np(1-p)}

These formulas emerge directly from the general formulas for sums of independent Bernoulli variables.

Bernoulli trial

For a Bernoulli variable ${Y}$ : ${\mu_Y = p}$ , ${\sigma_Y^2 = p(1-p)}$

The binomial variable

{X = Y_1 + Y_2 + \cdots + Y_n}

(sum of

{n}

independent Bernoullis):

\mu_X = \sum \mu_{Y_i} = np

\sigma_X^2 = \sum \sigma_{Y_i}^2 = n \cdot p(1-p)

Example: ${X \sim B(50, 0.6)}$ : ${\mu_X = 50(0.6) = 30}$ , ${\sigma_X = \sqrt{50(0.6)(0.4)} = \sqrt{12} \approx 3.46}$

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