About intuition of marginal PDF of two random variables.

Question

Let $f_{X,Y}(x,y)=\frac{3}{16}xy^2$, $0 \leq x \leq 2$, $0 \leq y \leq 2$. be the joint PDF of X and Y. Find the marginal PDF of $f_X$ and $f_Y$.

Question: I got $f_X(x)=\frac{1}{2}x$ and $f_Y(y)=\frac{3}{8} y^2$. My question is what does marginal PDF represent intuitively?

Answer 1

$f_X$ is the PDF of $X$. The joint PDF $f_{X,Y}$ gives you information about the randomness of both $X$ and $Y$ (as well as how they interact with each other); from this information, you can "throw away" the information about $Y$ (by integrating) so that you are left with only the information about the randomness of $X$, encapsulated in $f_X$.