I have a homework question about random vectors I wish to solve:
Choose a point randomly on the interval $[0,1]$ and label it $X_{1}$. Then choose a point randomly on the interval $[0,X_{1}]$ and label it $X_{2}$.Finally, choose a point randomly on the interval $[0,X_{2}]$ and label it $X_{3}$. Find the joint density function $f(x_{1},x_{2},x_{3})$ of the random vector$(X_{1},X_{2},X_{3})$ Then find the marginal density of $X_{3}$.
I know that by definition $F(x_{1},x_{2},x_{3})=P(X_{1}\leq x_{1},X_{2}\leq x_{2},X_{3}\leq x_{3})$.
I need help getting started, the problem I have is that $X_{2}$ is dependent on $X_{1}$ and $X_{3}$ is dependent on both $X_{1},X_{2}$.
I think this have to do with some conditional probability distribution function (I am probably miss translating this term), but I think that I only know it for two random variables $F_{Y|X}(y|x)=P(Y\leq y|X\leq x)$, is there something similar for $3$ random variables ?
Can someone please help me understand how to get started with this type of question ?