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I have a r.v. $X$~Poisson($\lambda$). Denote $Y=⌊X/a⌋$. What will be the distribution of $Y$ and how to calculate its mean? (A relaxation can be made if it makes the problem simpler: $a$ is an integer).

What I got so far: $P(Y=k)=P(X∈[ak,ak+a))=Fx(⌊ak+a⌋)-Fx(⌈ak⌉),k=0,1,2,... $, where $Fx$ is the CDF of r.v. $X$.

I can't go any further and derive a closed-form expression for $E[Y]$.

Any help will be much appreciated. Thank you.

Allan
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  • Is this for a class? If not, how does it arise? – Glen_b Aug 16 '17 at 10:57
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    Hi Glen. This is not from class, this is actually a part of the equation that I'm going to derive for my research topic. The scenario is that: packets arrive at a buffer according to Poisson arrival, the number of packets ahead of the one of interest is X~Poisson(lambda). The server can complete 'a' jobs at each visit. When my particular packet arrived, how many server visits are needed before my packet can be transmitted? (the server visit when my packet will be served doesn't account） – Allan Aug 16 '17 at 12:20
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    I derived an answer for a=2 as I used the even and odd terms in Taylor series. Do you know the equation of sum of every third, forth, fifth,... terms in Taylor series? These will definitely help me with my problem. – Allan Aug 16 '17 at 12:22
  • Allan, that's sometimes called "decimation" of a series. My answer to the duplicate question presents a general technique to carry it out. – whuber Aug 16 '17 at 14:37
  • I had a glance of that post but didn't realize the answer is exactly what I am looking for. Many thanks whuber! – Allan Aug 17 '17 at 08:07

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