Suppose we have a series of independent trials or repetitions and on each repetition or trial the probability of success ‘p’ remains the same. Then the probability that there are x failures preceding the first success is given by qx p.
A random variable X is said to have a geometric distribution if it assumes only non-negative values and its probability mass function is given by
P(X=x) = 0, otherwise
The geometric distribution is said to lack memory in a certain sense. Suppose an event E can occur at one of the times t=0,1,2,……and the occurrence(waiting) time X has a geometric distribution.
Thus, P(X=t) = qt .p; t = 0,1,2,…..
Suppose we know that the event E has not occurred before k, i.e., X ≥ k. Ley Y = X-k. Thus Y is the amount of additional time needed for E to occur. We can show that
P(Y=t/X ≥ k) = P9X = t) = pqt
Which implies that the additional time to wait has the same distribution as initial time to wait.
Since the distribution does not depend upon k, it, in a sense, ‘lacks memory’ of how much we shifted the time origin. If ‘B’ were waiting for the even E and is relieved by ‘C’ immediately before time k, then the waiting time distribution of ‘C’ is the same as that of ‘B’.
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