Marginal pdf.

Alright, I mentioned continuing on with the moment generating function, but I realized that I need to cover something before I can get really in depth with that. What we have to cover now are marginal PDFs. (I rarely ever cover discrete examples, since most problems are meant for continuous cases. It shouldn't matter, since most of the proofs transfer easily to summation with no problem).

Alright, you know the drill with one variable functions. For instance, what's the mean of the PDF $g(X)$? Well, it's:

$$\mu=\int_{S} xg(x)dx$$

Where $S$ is the support of $X$. Well, That's no problem. But what about two variables? Now we have a PDF that looks like $g(X_{1},X_{2})$. Here's where "marginal" PDFs come in. We have two variables to integrate over. So, the new expectation function (for whatever function of $X_{1}$ and $X_{2}$ you'd like, I'll use $f(X_{1},X_{2})$) looks like:

$$\int_{S_{2}}\int_{S_{1}} f(X_{1},X_{2})g(X_{1},X_{2})dx_{1}dx_{2}$$

Where $S_{1}$ and $S_{2}$ are the supports of $X_{1}$ and $X_{2}$, respectively. Now, we can do something cool here. Since this is the joint PDF of both variables, what happens when we sum over the support of one? Take, for example:

$$\int_{S_{1}} f(X_{1},X_{2})dx_{1}$$

Well, what does that give us? Keep in mind this effectively eliminates the variable $X_{1}$ out of the equation, so all that's left is variable $X_{2}$ in the PDF. Therefore, the probability only depends on the value of $X_{2}$. This makes it the marginal PDF. So, if we have a function like this:

$$\int_{B}\int_{S_{1}}X_{2}g(X_{1},X_{2})dx_{1}dx_{2}$$

Where $S_{1}$ is the entire support of $X_{1}$. Well, we can pull the $X_{2}$ out of the first integral since it only integrates over $X_{1}$, and that gives us:

$$\int_{B}X_{2}\left[\int_{S_{1}}g(X_{1},X_{2})dx_{1}\right]dx_{2}$$

The integral in the inner brackets is exactly the marginal PDF of $X_{2}$, so we'll denote that by $f_{2}(X_{2})$. Plugging that in gives us:

 $$\int_{B}X_{2}f_{2}(X_{2})dx_{2}$$

Which is just the mean of  $X_{2}$. You can also play around with more variables, and perhaps prove to yourself that the expectation of more than one variables is linear. As for now this should be good enough to finish my MGF post.