# Discrete Random Variables: The binomial distribution

## Discrete Random Variables

• A discrete random variable is a variable linked to the outcomes of a random event, where the variable can only take on a finite or countable number of values.
• Each outcome associated with a discrete random variable has a specific probability assigned to it.

## The Binomial Distribution

• The binomial distribution is a specific type of discrete probability distribution.
• It describes the likelihood of having exactly k successes in n Bernoulli trials.
• A Bernoulli trial is an experiment that results in a success with probability p and a failure with probability 1 - p.

## Characteristics of a Binomial Experiment

• The experiment has a fixed number of trials (n).
• Each trial should be independent of the others.
• There are just two possible outcomes for each trial, commonly termed success and failure.
• The probability of success (p) is the same for each trial.

## Parameters of the Binomial Distribution

• A binomial distribution is denoted as B(n, p), where n is the number of trials and p is the probability of success in a single trial.
• The expected value (mean) of a binomial distribution is np.
• The variance is np(1 - p). The standard deviation is the square root of the variance.

## Application of the Binomial Distribution

• Binomial distributions are extensively used in probability theory and statistics, particularly in situations where only yes-or-no questions are involved.

## The Binomial Theorem

• The binomial theorem describes the algebraic expansion of powers of a binomial. It is given by the formula (x+y)ⁿ= Σ [n!/(k!(n-k)!) xⁿ⁻ᵏ yᵏ].
• In relation to the binomial distribution, the binomial theorem gives the probabilities of the different outcomes.

## Binomial Coefficient

• Binomial coefficient or combination, represented by “n choose k”, gives the number of ways, disregarding order, that k items can be selected from n items.

## Probability Mass Function (PMF)

• In the context of the binomial distribution, the probability mass function gives the probability of getting a specific number of successes. It is given by P(X=k)= [n!/(k!(n-k)!)] pᵏ (1-p)ⁿ⁻ᵏ.
• The PMF is used to calculate the likelihood of each possible number of successes.