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Can someone explain to me why the mass formula for Poisson is $P(X=x)=\cfrac{e^{-\lambda}\lambda^x}{x!}$? Also why the expected value is $E(X)=\lambda$ and Variance is $Var(X) = \lambda$?

My book only gives the formula for mass and no explanation. It also gives the expected value formula and variance formula without any explanation. Can someone explain to me where these formulas come from so that I can better understand them?

Thank You!

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"My book only gives the formula for mass and no explanation." The Poison distribution is defined by the pmf you mentioned.

The expectation and variance are calculated using the pmf and definitions. (There are other methods if you know what the characteristic function is.)

The calculations are very standard materials. See for instance this set of notes:

https://llc.stat.purdue.edu/2014/41600/notes/prob1804.pdf

  • Forgive me, but when something is "defined" in mathematics. Does that mean that there is no proof or derivation or explanation associated with such a formula? Does "defined" mean, that's just how it is by the meaning of "defined" English-wise? –  Jan 13 '21 at 20:03
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    @BillBillwater: you can take a look at this: https://math.stackexchange.com/questions/50607/definition-of-definition Simply speaking, a "definition" means we decide an agreement on what we are talking about. For instance, we define "even numbers" as integers that are multiples of $2$. –  Jan 13 '21 at 20:06