# #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?

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