2.11 Parameters for a Binomial Distribution

Mean and Standard Deviation of a Binomial

For a binomial random variable X with parameters n (number of trials) and p (probability of success), the mean is μ = np, and the standard deviation is σ = √(np(1-p)). The mean tells you the expected number of successes over many repetitions, while the standard deviation measures the typical spread around that expected value.

Why These Formulas Work

A binomial variable is the sum of n independent Bernoulli trials, each with mean p and variance p(1-p). Since means and variances add for independent variables, the total mean is np and total variance is np(1-p), giving σ = √(np(1-p)). This connects binomial parameters back to basic rules for combining random variables.

Using the Parameters

These formulas let you quickly find the expected count of successes and how much that count varies from sample to sample, without listing the entire probability distribution. Larger n generally increases both spread (σ) and expected successes (μ), while p closer to 0.5 maximizes variability for a given n.

Key terms

Mean of a binomial random variable

μ = np, where n is the number of trials and p is the probability of success.

Standard deviation of a binomial random variable

σ = √(np(1-p)).

Variance of a binomial random variable

Var(X) = np(1-p).

Why μ = np works

A binomial variable is the sum of n independent Bernoulli trials, each with mean p; means add, so total mean = np.

Effect of p on variability

For fixed n, variability (σ) is greatest when p = 0.5 and smallest when p is near 0 or 1.

Try an AP-style question

X is binomial with n = 50 and p = 0.2. What is its mean?

1. 8
2. 40
3. 10
4. 0.2