Modelling a random variable with the Binomial Distribution

A Bernoulli random variable is used to represent the 'number of successes' observed from a single observation of a process with two outcomes: 'success' or 'failure'. However, we noted that this model was limited in it's application because the number of successes could be either $0$ or $1$ .

A binomial random variable extends upon this concept and represents the number of successes observed across observations of multiple Bernoulli random variables that have identical probability distributions and whose outcomes are independent of each other.

The binomial distribution is a probability distribution which describes the models a Bernoulli random variable $X$ . Like the Bernoulli distribution, the binomial distribution has a parameter $p$ which describes the probability of success for a single observation of the underlying Bernoulli random variable. It also introduces a new parameter $n$ which describes the number of observations which are taken. The formula for the binomial distribution is given by

$P(X = x) = {}^nC_x p^x(1-p)^{n-x}$

Where we recall that

${}^nC_x = \frac{n!}{(n-x)!x!}$

The mean and variance of the binomial distribution are given by:

$E[X] = np$ and $\operatorname{Var}(X) = np(1-p)$

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