Joint density function of X and X+Y, standard normal random variables Let $X$ and $Y$ be independent standard normal random variables. Find the joint density function of $X$ and $X+Y$.
My attempt: $P(X=x \cap X+Y=z)=P(X=x \cap X=z-Y)=P(X=x \cap Y=z-x)=$  ?
As you can see, I'm getting very far that way. I do know the marginal density functions of $X$ and $Y$. Namely, $$f_X(x)=P(X=x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$$ and $$f_Y(y)=P(Y=y)=\frac{1}{\sqrt{2\pi}}e^{-y^2/2}$$
So once I have the problem set up correctly, I should be able to use these formulas. Please let me know if you can help! Thanks!
Edit:
Perhaps the easiest way to find this is with the formula: $f_{X,X+Y}(x,z)=f_{X\mid X+Y}(x\mid z)*f_{X+Y}(z)$. I think I can find $f_{X+Y}(z)$ but how do I find $f_{X\mid X+Y}(x\mid z)=P(X=x \mid X+Y=z)$?
 A: Your approach didn't work out as the random variables you are dealing with are continuous, and hence $P(X=x)=0$. And I think you need to assume $X$ and $Y$ are independent random variables for this problem.
What you can do is, that, if $X$ and $Y$are independent, you can find the joint density of $X$ and $Y$, as,
$$f_{X,Y}(x,y)=f_X(x)f_Y(y)$$
Then consider the transformation $u:\mathbb{R}^2\rightarrow\mathbb{R}^2$ as 
$$u(x,y)=(x,x+y)$$
Find out the Jacobian of the transformation and multiply it with $f_{X,Y}(x,y)$ to get the joint density of $X$ and $X+Y$.
For more on change of random variables see here.
A: Since $X$ and $Y$ are independent standard Normals, their joint pdf, say, $f(x,y)$ is simply the product of the individual pdfs:

Then, we seek the transformation:

where the Transform function (from the mathStatica package) automates the requested transformation using the Method of Transformations. The domain of support is $R^2$. All done. 
If you would like to carry out the steps manually, and are not familiar with the Method of Transformations, see, for instance, Section 4.2 A of Chapter 4 of our book, "Mathematical Statistics with Mathematica". A free download of the chapter is available here:
http://www.mathStatica.com/book/Rose_and_Smith_2002edition_Chapter4.pdf
Just for fun, here is a 3D plot of the theoretical joint pdf just obtained, and superimposed on top, a Monte Carlo simulation of the same $(X,X+Y)$ transformation:

A: Simply use bivariate density formula as $x$ and $x+y$ are just two zero mean Gaussian RVs with correlation coefficient $\frac{1}{\sqrt2}$.
