Key attributes of a binomially distributed random variable

1. The experiment consists of \(n\) repeated trials.
2. The repeated trials are independent.
3. Each trial has two possible outcomes: either a 'success' or a 'failure'.
4. The probability of success is the same in each trial.

We define a binomial experiment as which possesses the above four properties.

Let \(X\) be the number of successes in \(n\) trials of a binomial experiment and let the probability of success be \(p\).

\(X\) is then called a binomial random variable.

Notation: \(X \sim B(n, p)\) where \(n\) and \(p\) are called the parameters of the distribution.

The probability distribution function (p.d.f.) of the binomial random variable \(X\) (which is the number of successes in \(n\) independent trials) is given by:

\[P(X=k) = {}^nC_k p^k q^{n-k} = {}^nC_k p^k (1-p)^{n-k}, \quad k=0,1,2, … n\]

The use of this formula is not required for exams