Get the marginal density from the joint density

Question

Hello ladies and gentlemen, I have a simple theoric issue to propose.

Given a random vector X$=(X_1,X_2,X_3)$ with joint density function $ρ(x_1,x_2,x_3)$, I know that

$\frac{\partial ^nF_X(x_1,...,x_n)}{\partial x_1...\partial x_n}=ρ(x_1,...,x_n)$,

now my question is, how can I get the single marginal density, for example $ρ_{x_2 }(x_2)$, without integrating? I mean there is a way that I can derivate the CDF some time and get the marginal density?

For example: $ρ_{x_2 }(x_2)=\frac{\partial ^2F_X(x_1,x_2,x_3)}{\partial x_1,\partial x_3}$ is it correct? Thanks to everyone who will help.

Answer 1

$\rho(x_2)=\frac{\partial}{\partial x_2}F_X(\infty,x_2,\infty)$

By definition $F_X(x_1,x_2,x_3)=P[X_1\leq x_1, \,X_2\leq x_2,\, X_3\leq x_3]$. What you want is $P[X_2\leq x_2]$. Since $X_1$ and $X_3$ are definitely less than $\infty$, the event $\{X_1\leq \infty, \,X_2\leq x_2,\, X_3\leq \infty\}$ is same as the event $\{X_2\leq x_2\}$. So, $F_X(\infty,x_2,\infty)=P[X_2\leq x_2]$. You can get the density of $X_2$ by differentiating: $\rho(x_2)=\frac{d}{dx_2}P[X_2\leq x_2].$