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I'm having issues proving the following identity:

$P(X ≥ j)$ = $P(Y ≤ t)$, where $X$~ Pois($\lambda$$t$) and $Y$ ~ Gamma($j$, $\lambda$)

More specifically, I can prove it algebraically but not with a story. I am getting mixed up with the switching of j and t. Does anyone have a story proof for this?

NOTE: A "proof" that does not involve much algebra but rather a story of a distribution. So, in this instance, the author of the post is looking for a story example that relates the Poisson Distribution and the Gamma Distribution

Dana K
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Joseph R
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    What's a story proof? I am not familiar with the term, so I expect it might be new to a number of people. – Glen_b Nov 08 '17 at 01:27
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1 Answers1

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I believe I can tell the "story". We have events of some sort (say, arrival of a customer) occurring according to a Poisson process with rate λ. X (it should actually be X(t)) is the number of events occurring up to time t - the number of customers arriving up to time t. Y is the sum of j Exponential random variables and is the time at which the j-th event happens, or in the example the time of arrival of the j-th customer. Clearly, saying that X(t) is greater than j is equivalent to saying that the j-th event occurred sometime before t. So the "stories" behind these two probabilities are the same - at least j customers arrived up to time t if and only if the arrival time of the j-th customer was earlier than t.

Zahava Kor
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  • This is my first time to hear the term "story proof" +1, Do your "story" means "sum of exponential" is gamma? – Deep North Nov 08 '17 at 02:17
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    Only the person who came up with this question knows what "story" means. It seems that any explanation that does not involve algebraic calculations is a "story". – Zahava Kor Nov 08 '17 at 18:45