# The geometric distribution

## Understanding the Geometric Distribution

• The geometric distribution represents the number of trials required for a first success in repeated independent Bernoulli trials.
• Each Bernoulli trial is independent and has only two possible outcomes, often labelled “success” and “failure”.
• The geometric distribution is therefore linked to events of the form ‘What is the probability that the first success occurs on the nth trial?’

## Parameters of the Geometric Distribution

• The geometric distribution has a single parameter, p, representing the probability of success on each trial.
• The parameter p must be a number between 0 and 1, inclusive, thus 0 ≤ p ≤1.

## Probability Mass Function (PMF)

• The PMF of a geometrically distributed random variable is given by P(X=n) = (1-p)^(n-1)p, where n is a positive integer (n=1, 2, 3, …)
• This function calculates the probability that the first successful outcome occurs on the nth trial.

## Cumulative Distribution Function (CDF)

• The CDF of a geometrically distributed random variable is given by F(n) = 1 - (1-p)^n
• This function calculates the probability that the first successful outcome occurs within the first n trials.

## Expectation and Variance of Geometric Distribution

• The expected value, or mean, of a geometrically distributed random variable is given by E(X) = 1/p. That means, on average, how many trials should be expected to occur before the first success.
• The variance of a geometrically distributed random variable is given by Var(X) = (1-p)/p^2. It describes the spread – the variability or dispersion – of the distribution.

## Memorylessness of Geometric Distributions

• Geometric distributions are memoryless. This means that the probability of seeing the first success in the next trial is always the same, regardless of how many failures have been seen so far.
• In other words, past outcomes do not affect the probabilities of future outcomes.

Remember practice is key. Work on problems involving the geometric distribution, such as finding the PMF, CDF, expected value, variance, or using the memoryless property.