conditional expectation value of sum of random variables

Question

let $v_1, v_2, ... v_n$ are random variables of uniform distribution on [-1,1]

And let $s=\sum_{i=1}^n v_i$

What is $E[s | s>0 ]$?

It means, what is the expectation value of s when s is bigger than $0$?

Answer 1

If you want the asymptotic expectation, you could use the symmetry of the underlying distributions, the central limit theorem, and the half-normal distribution to say that $E[s \mid s>0 ] = E[|s|]$ and that $\frac1{\sqrt{n}} s$ converges in distribution to $N(0,\frac13)$, leading to $$\frac1{\sqrt{n}}E[s \mid s>0 ] = \frac1{\sqrt{n}}E[|s|] \to \sqrt{\frac{2}{3\pi}}\approx 0.46$$ as $n$ increases

As a simulation in R to confirm this:

n <- 100
set.seed(1)
cases <- 10^6
vmat <- matrix(runif(cases*n, -1, 1), ncol=n)
s <- rowSums(vmat)
mean(s[s>0])
# 4.607967
mean(abs(s))
# 4.609709
sqrt(n) * sqrt(2/3/pi)
# 4.606589

This is not exact for small $n$: for example for $n=1$ it suggests about $0.46$ rather than the correct $\frac12$, and $n=2$ it suggests about $0.65$ rather than the correct $\frac23$