explain why memoryless property does not apply to the binomial random variable

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Expert Solution

The condition Y>a is fundamentally different in the special case of the Geometric distribution than in a Negative Binomial distribution with r>1. In a Geometric distribution, this condition immediately tells us there has been no successes by trial aa since r=1. While in the latter, it is possible that we have had some successes so long as there has been less than r of them. The condition doesn't itself tell us how many successes we've had by stage aa so it's impossible to reduce it to P(Y>b) since we cannot conclude that there have been no successes so far.

Since this is the case, of course, the memoryless property does not hold as the condition clearly affects the probability of reaching r successes by some number of trials if we've already had a few.

On the other hand, for the Geometric distribution if Y>a then since r=1 this means there have been no successes, and the memoryless property follows from independence and the constant probability of success.

orchestra answered 1 year ago

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