# How to get from joint distribution F(x,y) to f(x,y) to calculate the marginal distribution of X?

I have this homework question I'm not 100% sure how to tackle.

I have a random vector with joint distribution function F(x,y) and am asked to find the marginal distribution function.

I think need to get to $fx(x,y)$, but I'm not sure how to do this.

My thinking is that I need to differentiate F(x,y) by both x and y, to get f(x,y) and then integrate over y, but this gets me the marginal density for x.

Any pointers would be greatly appreciated

EDIT - Thanks Glen. I've switched to my proper user id. It's late here, I'll revisit tomorrow.

EDIT2 - to clarify, the first q in the homework starts at $F(x,y)$ and asks for marginal probability. The second starts with $f(x,y)$ and asks for marginal density so I'm assuming that in the first instance, they're after a pdf, and in the second, a cdf; to rule out suspicions of casual terminology use.

I need to differentiate F(x,y) by both x and y, to get f(x,y) and then integrate over y,

So after you do step 1 (differentiate over $x$), you have some object.

You then (step 2) differentiate with respect to $y$, and immediately (step 3) integrate it again with respect to $y$. What is the effect of doing both of those things?

Hint: consider instead the limit of $F(x,y)$ as $y\rightarrow \infty$

• Thanks for the reply Glen. From your question - differentiating and integrating cancel each other out. If I slip an x or y into the integration then I get to a density, not a distribution as such - I think this is the case, but this is one of my confusion loci. – ST. Mar 22 '13 at 4:58
• Yes, but the first step is just to avoid doing something useless, eh? The we can set about working out what you really want to do – Glen_b Mar 22 '13 at 6:30
• Thanks Glen - I see what you're saying and had thoughts along those lines, thinking I can simply differentiate by x (as the F(x,y) is already summed across y) to get to a marginal density for x... and unsure where to go from there. So i'm now thinking I need to differentiate again by x, to get to a distribution. – ST. Mar 22 '13 at 6:47
• Differentiation won't take you from density to distribution. – Glen_b Mar 22 '13 at 7:44
• I think I might have it now - take F(x,y), differentiate by x and y, then integrate y but instead of using -∞ -> y, use y -> ∞. – ST. Mar 23 '13 at 6:16

If by distribution you really do mean cumulative probability distribution function, then note that for each real number $x$, $$F_X(x) = \lim_{y \to \infty} F_{X,Y}(x,y)$$ which is colloquially written as $F_X(x) = F_{X,Y}(x,\infty)$. On the other hand, if your instructor is being lax and writing "find the marginal distribution" when what he really wants is the marginal density, then proceed as above and then get $f_X(x)$ by differentiating $F_X(x)$ that you just found.