How is the mean of a geometric distribution calculated?

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The mean of a geometric distribution is calculated using the formula 1/p, where p represents the probability of success in each individual trial. This is because a geometric distribution models the number of trials needed until the first success occurs.

For instance, if the probability of success on each trial (p) is known, the expected number of trials (mean) until the first success can be computed as the inverse of that probability. If p is high, the expected number of trials will be lower, reflecting a higher likelihood of achieving success quickly. Conversely, if p is low, the mean will be higher, indicating that more trials are expected before the first success is achieved.

This understanding is critical in applications such as reliability testing and quality control, where determining how many attempts are typical before a desired outcome occurs is essential.