intuitive proofs can be formal too

[gusl]

Let N ~ Poisson with mean mu.
Let X|N=n ~ Binomial(n,p).

Show that X ~ Poisson with mean p*mu.

source: STAT560 - Assignment 2, Problem 2

What most people call a "formal" proof (and what appears in the solution linked) involves writing down the relevant pdfs and working out infinite sums.

I prefer to prove this by simple intuitive logic:

"N ~ Poisson with mean mu" means that N is a process of people arriving, e.g. in an hour, an average of mu people will arrive.

"X|N=n ~ Binomial(n,p)" means that there will be p females per person arriving (regardless of how many people arrive).

Thus X ~ Binomial(N,p) is the process of females arriving.
The mean of X, therefore, is the average number of females arriving in an hour.
The conclusion follows trivially.
QED.

I find this a better proof: simpler and more intuitive. IMHO, it's just as formal as the proof involving infinite sums, except perhaps for the fact that it uses a non-standard (and possibly tacit) formalism.

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