Given two barrels of water, $A,B$, with 1 liter each. We pour an $X\sim U[0,1]$ amount of water from $A$ to $B$ and then $Y$ amount of water randomly from $B$ to $A$ $(Y|X=x\sim U[0,1+x])$.

Calculate the CDF of $Z$ - the amount of water in barrel A after both transfers.

My try:

Obviously $Z=1-X+Y$, so for $t\in[0,2]$, we calculate (using law of total probability):

$F_Z(t)=\mathbb{P}(Z\le t)=\mathbb{P}(1-X+Y\le t)=\int_0^1\mathbb{P}(Y\le t+x-1|X=x)f_X(x)=\int_0^1\mathbb{P}(Y\le t+x-1|X=x)$

How can I determine the limits of integration given different values of $t$?

Any help would be appreciated


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.