0
$\begingroup$

Let $X$ and $Y$ by positive independent random variables. Let $f(x,y)=\frac{ax}{y^2+ay}-\frac{ab}{y}$ where $a>0$ and $b>0$ are constants.

I am wondering if the following is true:

$$\begin{align}P[f(X,Y)\leq T]&=P[f(X,Y)\leq T~|~X\leq cY]P[X\leq cY]+P[f(X,Y)\leq T~|~X> cY]P[X> cY]\\ &\geq P\left[\frac{c}{Y+a}-\frac{ab}{Y}\leq T\right]P[X\leq cY].\end{align}$$

I think the equality follows by the law of total probability, though I am not sure if it applies here, given that I dealing with two random variables... does their independence play a role?

The inequality is due to the fact that, in the conditioned expression, $X$ is at most $cY$, that $f(x,y)$ is increasing in $x$, and that $Y>0$.

$\endgroup$
1
$\begingroup$

Your expression holds, and would hold even if the variables were not independent.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.