Probability Conditional Expectation

[ctownballer03]

Suppose X and Y are independent Poisson random variables with respective parameters λ and 2λ.
Find E[Y − X|X + Y = 10]3: I had my Applied Probability Midterm today and this question was on it. The class is only 14 people and no one I talked to did it correctly. The prof sent out an e-mail saying how no one did it correctly and we need to work on it and get it figured out and corrected by our next class and frankly I'm still super stuck (my professor is pretty useless, I can't utilize him as resource for anything in this class). Anyways, this is my attempt at doing this however I realize that I've made a mistake and even though X and Y are independent, Y=y and X+Y=10 are NOT independent events so the cancellation that I did is not a legal move.. Any other ideas of how to approach this problem, I feel like I'm back to square one and I'm not sure where to go. Thank you!

 

[Ray Vickson]

Suppose X and Y are independent Poisson random variables with respective parameters λ and 2λ.
Find E[Y − X|X + Y = 10]3: I had my Applied Probability Midterm today and this question was on it. The class is only 14 people and no one I talked to did it correctly. The prof sent out an e-mail saying how no one did it correctly and we need to work on it and get it figured out and corrected by our next class and frankly I'm still super stuck (my professor is pretty useless, I can't utilize him as resource for anything in this class). Anyways, this is my attempt at doing this however I realize that I've made a mistake and even though X and Y are independent, Y=y and X+Y=10 are NOT independent events so the cancellation that I did is not a legal move.. Any other ideas of how to approach this problem, I feel like I'm back to square one and I'm not sure where to go. Thank you!

If $X \sim \text{Po}(a)$ and $Y \sim \text{Po}(b)$, what is $P(X = k | X + Y = n)$?

Hint: $(X|X+Y=n)$ is a familiar discrete random variable, whose values range from $0$ to $n$.

 

[ctownballer03]

If $X \sim \text{Po}(a)$ and $Y \sim \text{Po}(b)$, what is $P(X = k | X + Y = n)$?

Hint: $(X|X+Y=n)$ is a familiar discrete random variable, whose values range from $0$ to $n$.

Things have just got interesting. I may have to post again later tonight if I get stuck but you have put me on the right track. So the PMF of X=k|X+Y=n where X+Y are poisson RVs is going to be a binomial.

 

[ctownballer03]

Solved it now.

Thank you for the hint.

 

[Ray Vickson]

Solved it now.

Thank you for the hint.

Just as a matter of interest: for independent $X_a \sim \text{Po}(a)$ and $X_b \sim \text{Po}(b)$, what did you obtain as the distribution of $(X_a|X_a+X_b=n)$?