This is a followup to the question in Bounding the Density of the Maximum of N Random Variables
I have a random variable, X, whose cdf is bounded as below:
$ \Pr \{X \le x \} \le \underset{i}{\prod} \Pr\left\{\xi_i \le x\right\} $,
How do I bound the density of $X$ ?
Can it be written as $f_X \le \sum_i f_{\xi_i} \prod_{j \ne i} \Pr\left\{\xi_j \le x\right\}$.
I am trying to write a proof for the above. Any hints ?