Two dots are drown on the real numbers axis. Their distances from 0 are random variables S.T ~Normal(1, 0.25) for each dot.
Q: What's the expected value of the distance between the two dots.
Final Answer: $1/\sqrt{\pi}$
My Approach:
Let's assign $X_1$ to first dot's distance from 0 and $X_2$ to second dot's distance from 0.
if $X_1>X_2$, then:
$E(Distance)=E(X_1-X_2)=E(X_1)-E(X_2)=1-1=0$
else:
$E(Distance)=E(X_2-X_1)=E(X_2)-E(X_1)=1-1=0$
In both cases we got 0, What did I do wrong?
Side Note: ~Normals refers to normal distribution.
We know that $X_1 - X_2 \sim \text{Normal}\left(0,\frac{1}{2}\right)$ (assuming the $0.25$ is a variance, not a standard deviation)
The distance between the points is $D:=|X_1-X_2|$ has a folded normal distribution with mean $\frac{1}{\sqrt{2}} \sqrt{\frac{2}{\pi}} = \frac{1}{\sqrt{\pi}}$.