What is the Distribution of Z When X and Y are Independent N(0, 1) Variables?

[johnny872005]

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

 

[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.

 

[johnny872005]

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:
answer: thefunc
reason: because x y and z

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

 

[Pere Callahan]

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.

 

[Pere Callahan]

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