Variance of sum

Question

Could you please tell me how to solve this problem?

Suppose $Y=\sum_i X_i$, where $X_i$ random variable and independent with pdf $f(x)$ Find the pdf of $Y$ and compute $E(Y)$ and $Var(Y)$.

Thanks a lot.

Answer 1

As Clement pointed out, the the sigmoid function is not a valid pdf because its integral is not equal to $1$. I am almost certain that you mean that it is the cdf, as it is a nondecreasing function that tends to $0$ and $1$ as $x$ tends to $-\infty$ and $\infty$ respectively.

Hints: In general, $E[Y]=E\left[\sum_i X_i\right] = \sum_i E[X_i]$. If the $X_i$ are also independent, then $\operatorname{Var}(Y) = \operatorname{Var}\left(\sum_i X_i\right) = \sum_i \operatorname{Var}(X_i)$.