Geometric distribution
The geometric distribution models the number of trials until the first success, when each trial has success chance p. P(X = k) = (1-p)^(k-1) x p. The mean is 1/p.
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Analogy
Rolling a die until your first six: how many rolls do you wait?
Example
p = 1/6P(first six on roll 3)= (5/6) x (5/6) x (1/6) = 25/216 = 0.116Mean wait = 1 / (1/6) = 6 rolls
In practice
Used for the number of attempts until a first sale, first defect or first pass.
Practice
Flashcards
- Mean number of trials until first success?1/p, from the geometric distribution.
Questions people ask
Why is the mean of a geometric distribution 1 divided by p?
A rare success takes longer to arrive. If p is one in six, you wait about six trials on average, since each trial gives one-sixth of a success. The result 1/p matches this intuition and can be proved by summing the series.
Does the geometric distribution have a memory?
No. If you have failed ten times already, the chance of success on the next trial is still p. The remaining wait has the same distribution as when you started. This memoryless property is why past failures do not make success due.