Expected value of X~Geom(p) given X+Y=z

[BerriesAndCream]

TL;DR Summary

finding a conditioned expected value

Hello everyone.

If X, Y are two independent geometric random variables of parameter p, and Z=X+Y, what's E[X|Z=z]?

I have calculated the distribution of P(Z=z) and I have then found that the conditional probability P(X=x|Z=z) equals 1/(z-1).
How can I now find the conditioned expected value?

 

[andrewkirk]

What is the value of $E[X|Z=z]+E[Y|Z=z]$?

Which is the bigger of $E[X|Z=z]$ and $E[Y|Z=z]$ (trick question).

 

[BerriesAndCream]

What is the value of $E[X|Z=z]+E[Y|Z=z]$?

Which is the bigger of $E[X|Z=z]$ and $E[Y|Z=z]$ (trick question).

1. E[X|Z=z] + E[Y|Z=z] = E[X+Y|Z=z] = E[Z|Z=z] = z?
2. Is there a way of knowing?

 

[BerriesAndCream]

Could it be that since P(X=x|Z=z) = 1/(z-1), which is an uniform distribution, the expected value I'm looking for can simply be found by using the formula for the expected value of the uniform distribution?

edit: I tried to do it that way, but I couldn't find the right answer. So i tried by using the formula for the expected value. What I did not realise at first is that I had to sum from x=1 to x=z-1, right? By doing so I get that E[X|Z=z]=
Σ_x=1^z-1 x·P(X=x|Z=z) =
Σ_x=1^z-1 x/(z-1) = 1/(z-1)·Σ_x=1^z-1 x =
1/(z-1) · (z-1)(z-1+1)/2 = z/2.

Is this valid?