Binomial probability distribution

Probability distribution of the number of successes in a sequence of independent experiments

Template:Probability distribution

The binomial probability distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is a fundamental concept in probability theory and statistics, often used in various fields such as medicine, biology, and social sciences to model binary outcomes.

Definition

A binomial distribution with parameters \( n \) and \( p \) is the discrete probability distribution of the number of successes in a sequence of \( n \) independent experiments, each asking a yes–no question, and each with its own boolean-valued outcome: success (with probability \( p \)) or failure (with probability \( 1-p \)).

The probability mass function (pmf) of a binomial distribution is given by:

\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]

where:

• \( \binom{n}{k} \) is a binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \).
• \( n \) is the number of trials.
• \( k \) is the number of successes.
• \( p \) is the probability of success on an individual trial.

Properties

Mean and Variance

The mean (expected value) of a binomial distribution is given by:

\[ \mu = np \]

The variance of a binomial distribution is given by:

\[ \sigma^2 = np(1-p) \]

Skewness and Kurtosis

The skewness of a binomial distribution is:

\[ \gamma_1 = \frac{1-2p}{\sqrt{np(1-p)}} \]

The kurtosis is:

\[ \gamma_2 = \frac{1-6p(1-p)}{np(1-p)} \]

Applications

The binomial distribution is widely used in various fields:

• In medicine, it can model the number of patients responding to a treatment out of a sample.
• In quality control, it can model the number of defective items in a batch.
• In genetics, it can model the inheritance of traits.

Related Distributions

• The Bernoulli distribution is a special case of the binomial distribution where \( n = 1 \).
• The Poisson distribution can be used as an approximation to the binomial distribution when \( n \) is large and \( p \) is small.
• The normal distribution can approximate the binomial distribution when \( n \) is large and \( p \) is not too close to 0 or 1, according to the central limit theorem.

Also see

• Bernoulli trial
• Probability distribution
• Poisson distribution
• Normal distribution
• Central limit theorem

References

• Feller, W. (1968). An Introduction to Probability Theory and Its Applications. Vol. 1. Wiley.
• Ross, S. (2014). A First Course in Probability. Pearson.