Section 47 Binomial Distribution

• For discrete data the Binomial distribution is one of the common distributions.

• A Binomial random variable with parameter \(n = 1\) is equivalent to a Bernoulli random variable, i.e. there is only one trial.

• A random experiment consists of \(n\) Bernoulli trials such that:

  • The trials are independent, i.e. the probability of success is equal to \(p\) in each trial

  • Each trial results in only two possible outcomes, labeled as success and failure

  • The probability of a success in each trial, denoted as \(p\), remains constant

• The random variable \(X\) that equals the number of trials that result in a success has a Binomial random variable with parameters \(0 < p < 1\) and \(n = 1, 2, 3, ...\)

• If the random variable \(X\) follows Binomial distribution with the parameters \(n=1,2,...\) and \(0<p<1\), then the probability mass function of \(X\) is:

\[ \large f(x; n, p) = \binom{n}{x} p^x(1-p)^{n-x} \]

• We say that \(X\) has a Binomial distribution, and denote this by writing: \(X \sim Bin(n, p)\)

• If \(X\) is a Binomial random variable with parameters \(p\) and \(n\):

\[ \large \mu = E(X) = np \]

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