Let $X$ be a continuous random variable with uniform probability distribution between $0$ and $S$, i.e, $X \sim U(0, S)$. Let Y be another continuous random variable distributed uniformly between $0$ and $X$, i.e, $Y \sim U(0, X)$.
- I want to know if there is a valid joint probability distribution.
- If my direction for caclulating the joint CDF as shown below is correct.
$F_{X,Y}(u_{o}, v_{o}) = P(X<u_{o}, Y<v_{o}) = \frac{P(Y<v_{o} | X < u_{o}) P(X < u_{o}) }{P(Y < v_{o})} $
$P(Y<v_{o} | X < u_{o}) = \int_{-\infty}^{u_{o}}{P(Y<v_{o} | X = u)du}$
$P(X < u_{o}) = \frac{u_{o}}{S}$
$P(Y < v_{o}) = \int_{-\infty}^{\infty}{P(Y < v_{o} | X = u)du}$
Best Answer
It is evident that $Y\leq X$ almost surely.
Consequently if $u_{0}\leq v_{0}$ then: $$P\left(X\leq u_{0},Y\leq v_{0}\right)=P\left(X\leq u_{0}\right)$$ and the RHS is easy to find.
If $0<v_{0}<u_{0}<S$ then:
$$\begin{aligned}P\left(X\leq u_{0},Y\leq v_{0}\right) & =P\left(X\leq v_{0},Y\leq v_{0}\right)+P\left(v_{0}<X\leq u_{0},Y\leq v_{0}\right)\\ & =P\left(X\leq v_{0}\right)+\int_{v_{0}}^{u_{0}}P\left(Y\leq v_{0}\mid X=x\right)f_{X}\left(x\right)dx\\ & =\frac{v_{0}}{S}+\frac{1}{S}\int_{v_{0}}^{u_{0}}P\left(Y\leq v_{0}\mid X=x\right)dx\\ & =\frac{v_{0}}{S}+\frac{1}{S}\int_{v_{0}}^{u_{0}}\frac{v_{0}}{x}dx\\ & =\frac{v_{0}}{S}+\frac{v_{0}}{S}\left[\ln x\right]_{v_{0}}^{u_{0}}\\ & =\frac{v_{0}}{S}+\frac{v_{0}}{S}\left(\ln u_{0}-\ln v_{0}\right) \end{aligned} $$
Personally I would go for the notation: $$P\left(X\leq x,Y\leq y\right)=\frac{y}{S}+\frac{y}{S}\left(\ln x-\ln y\right)$$where $0<y<x<S$.