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 https://math.stackexchange.com/q/12068/4051. The readers would take a look at it too.

**Contents**hide

#### Best Answer

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).

### Similar Posts:

- Solved – Expected log value of noncentral exponential distribution
- Solved – Bias of the maximum likelihood estimator of an exponential distribution
- Solved – Update gamma prior with new rate parameter instead of observations
- Solved – the mean of this exponential random variable
- Solved – Square of gamma random variable