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"The number of events in a month is distributed as a Poisson random variable with mean 2. What is the probability that there are 30 events in a year?" (Treat months as equal length)

I have the above question in a text book, so X~Poi(2). And I need to find out P(12X=30) ? The answer says that Y ~ Poi(24), (I'm guessing Y=12X) but it doesn't say why.

Could someone please explain or prove this result (regarding Y) to me?

Barok
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  • Why would you look at $12X$? You aren't assured of replicating a single month $12$ times. As to the official answer, that's how Poisson works. It scales with time...so it stays Poisson as you change the time window and the average scales as you'd think it would. – lulu Jan 08 '23 at 16:09
  • The problem does not say anything about different months being independent, so, $Y=12X$ is a possibility if we do not have independence (but that would change the answer since $P[12X=2]=0$). Most likely you are supposed to assume independence and use the fact that the sum of mutually independent Poisson variables is again Poisson. – Michael Jan 08 '23 at 16:12
  • If $X\sim\text{Poisson}(\lambda)$ and $Y\sim\text{Poisson}(\mu)$ then - if $X$ and $Y$ are independent - $X+Y\sim\text{Poisson}(\lambda+\mu)$. Exploit this nice property. – drhab Jan 08 '23 at 16:19
  • @drhab Thanks I understand the answer now using this property. Is there a proof for it? – Barok Jan 08 '23 at 16:45
  • See here for instance. – drhab Jan 08 '23 at 16:58

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