If I have a joint pdf of multiple random variables, say 3 for simplicity, $f_{X,Y,Z}(x,y,z)$, is it true that the marginal density functions derived from that joint probability distribution ( $f_{X}(x)$, $f_{Y}(y)$, $f_{Z}(z)$ ) are not guaranteed to be valid probability density functions (integrate to 1) in all cases. I can see that the pdf condition holds for the marginals when the random variables are independent but not quite sure what happens when we introduce dependence between them.
Do marginal density functions derived from a joint pdf always integrate to 1 (are they valid pdf's)?
$\begingroup$
$\endgroup$
3
-
4$\begingroup$ By definition, a marginal distribution describes a subset of the random variables and must be a valid distribution. If you do a calculation that violates that condition, then you have erred. $\endgroup$– whuber ♦Commented Dec 8, 2023 at 17:15
-
$\begingroup$ I just realized that my claim is equivallent to saying that the volume under the joint distribution is not 1. So dumb... $\endgroup$– StatisticoolCommented Dec 8, 2023 at 17:29
-
$\begingroup$ @Statisticool For closure purposes, you may want to well answer it yourself and accept your own answer :). $\endgroup$– AvrahamCommented Dec 8, 2023 at 17:30
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
1
Using two properties of the joint pdf:
$\int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} f(x, y, z) \, dx \, dy \, dz = 1$,
and $f_X(x) = \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} f(x, y, z) \, dy \, dz$
It follows that $\int\limits_{-\infty}^{\infty} f_X(x) \, dx = 1$
-
3$\begingroup$ Notice this assumes Fubini's Theorem applies. But you don't need it: the marginal density is that of the variable $X,$ namely, the one obtained by forgetting about all the other variables. It's that simple! $\endgroup$– whuber ♦Commented Dec 8, 2023 at 18:06