# Ind rand variable distribution

## Main Question or Discussion Point

Hi Guys.

I'm trying to understand how to solve the following problem. Any help and explanation would be greatly appreciated!
thanks.

Let X and Y be independent N(0, 1) random variables and let Z = X + Y.

What is the distribution of Z? Write down the density function of Z.
Also:
Show that E[Z|X > 0, Y > 0] = 2

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mathman
Have you learned about characteristic functions(cf)? If so then you get the cf of Z is the product of the cf's of X and Y. The cf of Z is then exp(-t2), so that Z is normally distributed with a variance of 2.

For the last question, set it up as a ratio of double integrals in x and y. Then convert to polar coordinates. It should be tedious, but workable.

ssd
If X~N(m1,s1^2) and Y~N(m2,s2^2) indeply, then
X(+-)Y ~ N(m1(+-)m2,s1^2+s2^2), it is easier to check from mgf or cf.

I'm still somewhat confused. What would you say as the final answer to the two questions I have above....?

If you could just list something like:
reason: because x y and z

Sorry if it seems bossy, I'm just really confused now after reading the above.
thanks

so that Z is normally distributed with a variance of 2.

It's hardly possible to state the answer to 1) more explicitly.

ssd
It's hardly possible to state the answer to 1) more explicitly.
No, you have to add 'and mean=0'. LOL.

O.k., you're right, of course.