The Geometric Distribution

• The Geometric Distribution is a type of probability distribution used to model the number of trials required to obtain the first ‘success’ in a sequence of Bernoulli trials, which are independent, identically distributed random variables.
• It’s considered a discrete distribution, meaning it measures a set of separate, distinct outcomes.
• The two parameters of a Geometric Distribution are p, the probability of success on a single trial, and k, the number of trials.
• Each trial in the Geometric Distribution is independent, meaning the outcome of one trial does not affect the outcome of another.
• The probability mass function (pmf) for a Geometric Distribution is given by the formula: P(X=k) = (1-p)^(k-1)*p, where k is the number of trials required to achieve the first success, and p is the probability of success on any given trial.
• The expected value or mean of a Geometric Distribution is 1/p, and its variance is (1-p)/p^2.
• The Geometric Distribution is memoryless, meaning that the probability of success does not change regardless of how many failures have occurred previously.
• You can use the Geometric Distribution to calculate probabilities in real-life situations where events are independent and the probability of success is constant - for instance, repeated flips of a fair coin.
• A key aspect in understanding the Geometric Distribution is the concept of ‘failure’. If the probability of ‘success’ is p, then the probability of ‘failure’ is 1-p.
• In a cumulative geometric distribution, the probability of obtaining the first success within a certain number of trials can be calculated by using the formula: P(X≤k) = 1 - (1-p)^k.
• The Geometric Distribution should not be confused with the Binomial Distribution, which describes the number of successes out of a fixed number of trials, whereas the Geometric Distribution describes the number of trials until the first success.