Homework Statement
Show that the integral of (1/x)sin(1/x)dx from 0 to 1 is converges absolutely?
Homework Equations
The Attempt at a Solution
Should we use the comparison test in this situation?
I have found that
E[X|Z=1] = E[X|X<Y] = 1/9
E[X|Z=0] = E[X|X>Y] = 8/9
by integrating, and conditioning on the random variable Y.
So E(X) = E(E(X|Z)) = (1/9)(1/3) + (8/9)(2/3) = 17/27,
which contradicts the fact that E(X) = 1, for X exponential with mean 1.
I am wondering where is...
Let X, Y be independent exponential random variables with means 1 and 2 respectively.
Let
Z = 1, if X < Y
Z = 0, otherwise
Find E(X|Z) and V(X|Z).
We should first find E(X|Z=z)
E(X|Z=z) = integral (from 0 to inf) of xf(x|z).
However, how do we find f(x|z) ?