# The binomial distribution

## Understanding The Binomial Distribution

• The Binomial Distribution is a specific type of discrete probability distribution.
• It models the number of successes in a fixed number of Bernoulli Trials or independent experiments with only two possible outcomes.
• Each Bernoulli trial is independent of each other and has the same probability of success, usually denoted by p.
• The distribution is denoted as B(n, p) where n is the number of trials and p is the probability of success on each trial.

## Properties of The Binomial Distribution

• The binomial distribution has two parameters: n (the number of trials) and p (the probability of success on each trial).
• The possible values of a Binomial random variable are the whole numbers from 0 to n.
• The mean of a binomial distribution B(n, p) is np, and the variance is np(1-p).

## Probability Mass Function (PMF) and Cumulative Distribution Function (CDF)

• The Probability Mass Function of a binomial distribution gives the probability of getting k successes in n trials. It can be calculated using the combination formula C(n, k) * p^k * (1-p)^(n-k).
• The Cumulative Distribution Function for a binomial distribution is the sum of the probabilities of the outcomes up to and including the defined upper limit.

## Working with The Binomial Distribution

• The binomial distribution is commonly used in statistical modelling and experiments where the outcome can be classified as a success or failure.
• The binomial coefficient or “n choose k”, denoted C(n,k) or nCk in many statistics textbooks, is an important concept when working with binomial distributions.
• As an example, the binomial distribution can be used to answer questions such as “what’s the probability of getting exactly 6 heads in 10 coin tosses?”
• The binomial distribution simplifies when p = 0.5 to a symmetrical distribution, commonly known as a bell curve.

To cement these concepts, review your lecture notes and work through as many practice problems as possible.