Expected Value, E[e^(x) | X+Y]?

Question

How do I show the expected value, of e^x?

Calculate E[e^(x) | X+Y]

Given gx(x) = gy(x) = e^((-x^2)/2) / (sqrt(2pi))
It is also known that X & Y are standard normal variables.

Thank you!

Answer 1

I'm not sure I can help on this one, partly because I'm not sure what the question is.

E[e^x | X+Y] is a random variable. Are you looking for the probability distribution function of this random variable or are you looking for the expected value of this random variable?

If the latter, then from what I can see:

1. There is the probability distribution of X+Y, which is just Gaussian.
http://en.wikipedia.org/wiki/Normal_distribution

2. Then there is the distribution of X|X+Y, which involves the calculation of conditional probabilities.
http://www.mathgoodies.com/lessons/vol6/conditional.html

In the equation P(A and B)/P(B), you'd have to use cumulative distribution functions and then take the derivative of the result.


3. Finally, there is the distribution of F(Z) where F = e^x and Z = X|X+Y
www.cis.rit.edu/class/simg713/Lectures/Lecture713-04-4.pdf
http://en.wikipedia.org/wiki/Random_variable

If this is not to be a tedious and meaningless exercise in algebra, I have to assume that the properties of the Gaussian and e^x interact to simplify the result.

Sorry I can't be of more help.