I 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?

 

[Gavran]

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?

Yes it is.
There is another approach as described by @andrewkirk in the post #2. The first question is answered in the post #3. What about the trick question? Is $E [X|Z=z]$ bigger than $E [Y|Z=z]$? or Is $E [Y|Z=z]$ bigger than $E [X|Z=z]$? or What is the third option?

 

[Gavran]

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?

Yes, it can be found by using a formula for calculating an expected value of a discrete uniform distribution.