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So, I have this problem and I am stuck on a sum. The problem I was given is the following:

The probability of a given number n of events (0 ≤ n < ∞) in a counting experiment per time (e.g. radioactive decay events per second) follows the function (Poisson distribution) P(n) = µ

a) show that ∞ ∑ n=0 P(n) = 1, i.e. that P(n) is normalised

b) compute the mean <n> = ∑

c) compute the standard deviation

I already did a) and b), where for b) I got <n>=μ. For c) I know that the variance is given by σ

I have no idea how to go from there in order to find <n

Thank you for your time.

So, I have this problem and I am stuck on a sum. The problem I was given is the following:

The probability of a given number n of events (0 ≤ n < ∞) in a counting experiment per time (e.g. radioactive decay events per second) follows the function (Poisson distribution) P(n) = µ

^{n}/n! *e^{-μ}, where µ is a real number. Recalling the series expansion of the exponential function e^{x}= ∑_{n=0}^{∞}x^{n}/n!a) show that ∞ ∑ n=0 P(n) = 1, i.e. that P(n) is normalised

b) compute the mean <n> = ∑

_{n=0}^{∞}nP(n)c) compute the standard deviation

I already did a) and b), where for b) I got <n>=μ. For c) I know that the variance is given by σ

^{2}=<n^{2}> - <n>^{2}. I know that <n>^{2}=μ^{2}but for <n^{2}>, which I know is given by ∑_{n=0}^{∞}n^{2}*P(n), I evolve until I get stuck with μe^{-μ}∑_{n=0}^{∞}(n*μ^{n-1})/(n-1)!I have no idea how to go from there in order to find <n

^{2}>, do you have any idea of how to go from there?Thank you for your time.

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