Solved – How to calculate the expectation of $left(sum_{i=1}^n {X_i} right)^2$

If $X_i$ is exponentially distributed $(i=1,…,n)$ with parameter $lambda$ and $X_i$'s are mutually independent, what is the expectation of

$$ left(sum_{i=1}^n {X_i} right)^2$$

in terms of $n$ and $lambda$ and possibly other constants?

Note: This question has gotten a mathematical answer on The readers would take a look at it too.

If $x_i sim Exp(lambda)$, then (under independence), $y = sum x_i sim Gamma(n, 1/lambda)$, so $y$ is gamma distributed (see wikipedia). So, we just need $E[y^2]$. Since $Var[y] = E[y^2] – E[y]^2$, we know that $E[y^2] = Var[y] + E[y]^2$. Therefore, $E[y^2] = n/lambda^2 + n^2/lambda^2 = n(1+n)/lambda^2$ (see wikipedia for the expectation and variance of the gamma distribution).

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