The geometric distribution is a probability distribution that models the number of failures before the first success in a series of independent Bernoulli trials. It is particularly useful in scenarios where you are interested in the waiting time until a specific event occurs. For example, it can be applied in various fields such as finance, sports, and commerce to analyze the likelihood of achieving a success after a certain number of failures ¹².

In mathematical terms, the geometric distribution is defined by a single parameter ( p ), which represents the probability of success on each trial. The probability mass function (PMF) gives the probability of observing ( y ) failures before the first success, expressed as ( P(Y = y) = (1-p)^y \cdot p ). The expected value and variance of the distribution can also be derived from this parameter, providing insights into the average number of trials needed to achieve the first success ²⁴.

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