$\min(S^3,T)$ with $S \perp T$

Question

Let $S$ and $T$ two random variables with exponential distribution of rate $\lambda$ and density $f(u)=\lambda e^{-\lambda u},u>0$. Find the density of:

• 1) $X=|S-T|$.

$\rightarrow X\sim Exp(\lambda)$

• 2) $Y=S^3$.

$\rightarrow f_Y(y)=\frac{1}{3}\lambda y^{-\frac{2}3{}}e^{-\lambda y^{\frac{1}{3}}}$

• 3) $Z=\min(S^3,T)$.

In this case I am having difficulty because, if I know that $S \perp T$, I don't know anything about the relationship between $S^3$ and $T$. Moreover, the graph doesn't help since $y=s^3$ is a function with a point of inflection in $(0,0)$. Can you help me?