User:IssaRice/Poisson distribution

Let X be the number of letters received in a day. If the letters arrive indenpendently blah blah blah, it should be clear that X has a Poisson distribution. Let's say you get about 4 letters on an average day. So it's a poission with lambda=4. But there is nothing special about using "day" as the interval size. What if we consider your mail on a yearly granularity. Then, you should get about 365*4 letters on an average year. Again, there is NOTHING SPECIAL about using "day" or "year" as the granularity here. So it should feel obvious that if Y is the number of letters received in a year, THAT should be a poisson distribution with lambda=365*4. But what is Y? Well, we could have X_1, ..., X_365 all be i.i.d possion(lambda=4). Then Y is simply the sum Y = X_1 + ... + X_365. What have we shown? We've shown that the sum of i.i.d poisson distributions is also poisson, and simply has the sum of the lambdas as the new lambda.

what if the possions have different distributions, but still are all independent? well, we could cut up our year into differently-sized chunks of time. so let X_1 be the letters received on the first day, let X_2 be the letters received in the following week, let X_3 be the letters received in the following 38 days, whatever. All the way up to say X_45 which is the last week. (for simplicity, i'm assuming all days are basically the same, e.g. you don't receive any more letters near your birthday or whatever...) Then on average, the mail you receive should be the sum of all the different lambdas. and we already established that the yearly mail distribution is poisson.

So "sum of possions = poisson with sum of lambdas" can be seen as an INVARIANCE THEOREM, stating that it doesn't matter how you cut up the intervals.

If you want the boring proof, this is pretty good. You should know what convolution means though.

BONUS QUESTION: how does the above not contradict the central limit theorem????