Discrete Random Variables: The geometric distribution

The geometric distribution is a probability distribution that describes the number of trials needed to get the first success in repeated independent Bernoulli trials.
Each trial is independent of the others and only two outcomes are possible—a success or a failure.
The geometric distribution is memoryless, like the exponential distribution, but for discrete rather than continuous random variables. This means that the probability of the first success being on the kth trial is the same regardless of how many failures have preceded it.

Characteristics of a Geometric Distribution

The geometric distribution is defined by a single parameter, p, which is the probability of success on any given trial.
The outcomes must take on non-negative integer values.
The distribution is right-skewed, with a shape depending on the success probability p.
The mean (expected value) is 1/p, and the variance is (1 - p) / p^2.

The geometric distribution is denoted by Geo(p), where p is the probability of success in a trial.
Expected value (mean) of a geometric distribution is 1/p.
The variance of a geometric distribution is (1 - p) / p^2 and the standard deviation is the square root of the variance.

For a geometric distribution, the probability mass function gives the probability of observing the first success on the kth trial.
It is given by P(X = k) = p(1 - p)^(k - 1) for k ≥ 1.

The cumulative distribution function for a geometric random variable gives the probability that the variable takes a value less than or equal to a certain value.
For a geometric distribution, it is given by F(k) = 1 - (1 - p)^k for k ≥ 1.

The geometric distribution is particularly useful for modelling situations where we are interested in the number of trials until the first success in binary outcome experiments.
Typical applications include modelling the number of attempts before passing a test, the number of coin tosses until the first head appears, or the number of calls made to reach a busy telephone line.