#StackBounty: #variance #random-variable #expected-value Expectation and variance of quotient of sums of positive, discrete, iid random…

Bounty: 50

Let ${X_i}{i=1}^n$ be $n$ positive, discrete (so positive integers) and IID random variables. Let ${c_i}{i=1}^n$ be constants and
$$Y=frac{sum c_iX_i}{big(sum X_ibig)^2} ; Z=frac{1}{sum X_i}$$

I’m trying to calculate $mathbb{E}[Y]$ and $text{var}(Y)$ in terms of $mathbb{E}[X_i]$‘s. Similarly for expectation and variance of $Z$. I’ve looked at other answers related to calculating the expectation of inverses and quotients, but they deal with more general cases and involve integrals and all.

Given the assumptions about $X_i$‘s that I listed out, how can $mathbb{E}[Y]$ and $text{var}(Y)$ be calculated?

Get this bounty!!!

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