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Let, \begin{equation} g(\alpha,\beta) = \begin{cases} \frac{\alpha}{\beta}, & \text{if } \alpha > \beta \\ 0, & \text{if } \alpha \leq \beta \end{cases} \end{equation} I want to find the CDF of $$Z=\max\left\{g(X,Y),g(X,C)\right\}$$ where$X,Y\geq0$ are independent random variables, and $C>0$ is a constant.

I am facing two difficulties:

  1. I could get the product of CDFs within $\max$ if two were independent. However, as they have $X$ in common, dependency is created. Therefore, I tried to condition on $X$ first and solve. However, in that case, $\frac{X}{C}$ becomes a constant, which is confusing me.
  2. I have more complications in applying the conditions of $X>Y$ and $X>C$ seperately to the analysis.

Can someone help me write the CDF of $Z$ in terms of the PDFs/CDFs of $X$ and $Y$?

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    $\begingroup$ Are $X$ and $Y$ independent? $\endgroup$
    – Zhanxiong
    Commented May 1, 2023 at 1:59
  • $\begingroup$ @Zhanxiong - yes $X$ and $Y$ are independent. $\endgroup$
    – Frey
    Commented May 1, 2023 at 2:01
  • $\begingroup$ Try to apply the cited theorem and mimic the calculation in stats.stackexchange.com/questions/606503/… $\endgroup$
    – Zhanxiong
    Commented May 1, 2023 at 2:06
  • $\begingroup$ I don't believe that you have completely specified what $Z$ is. What is $Z$ when $X < Y$ or when $X< C$? Since $X$ and $Y$ are independent nonnegative random variables, the event $(X < Y)$ has the same probability as the event $(X > Y)$. So, you need to specify what $Z$ is not just (i) when $X>Y$ and $X>C$ (as you have done), but also (ii) when $X\leq Y$ and $X>C$, (iii) when $X\leq Y$ and $X\leq C$, and (iv) when $X>Y$ and $X\leq C$. $\endgroup$
    – Dilip Sarwate
    Commented May 1, 2023 at 2:44
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    $\begingroup$ $Z$ has a point mass in $0$, so you need to distinguish $Z=0$ from $Z>0$, the computation of $$\mathbb P(Z>0, g(X/Y)<z,g(X/C)<z)$$ depending on the cdfs of $X$ and $Y$. $\endgroup$
    – Xi'an
    Commented May 1, 2023 at 9:18

1 Answer 1

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There are six different regions (in the first quadrant of the $x$-$y$ plane) to consider depending on the values of $X, Y, C$.

  • If $Y > C > X$, then $Z=\max\left\{g(X,Y),g(X,C)\right\} = \max\left\{0, 0 \right\} = 0.$
  • If $Y > X > C$, then $Z=\max\left\{g(X,Y),g(X,C)\right\} = \max\left\{0, \frac XC \right\} = \frac XC.$
  • If $X > Y > C$, then $Z=\max\left\{g(X,Y),g(X,C)\right\} = \max\left\{\frac XY, \frac XC \right\} = \frac XC.$
  • If $X > C > Y$, then $Z=\max\left\{g(X,Y),g(X,C)\right\} = \max\left\{\frac XY, \frac XC \right\} = \frac XY.$
  • If $C > X > Y$, then $Z=\max\left\{g(X,Y),g(X,C)\right\} = \max\left\{\frac XY, 0 \right\} = \frac XY.$
  • If $C > Y > X$, then $Z=\max\left\{g(X,Y),g(X,C)\right\} = \max\left\{0,0 \right\} = 0.$

Can you take it from here?

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