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 $Y>0$.


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


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