Question

Given two independent random variables $X_1$ and $X_2$ and a function and given that $P\Big[f\big(\mathbb{E}[X_i]\big)\leq\alpha\Big]\geq\beta$ , does the following inequality hold?

$P\Big[\max \Big\{f\big(\mathbb{E}[X_1]\big),f\big(\mathbb{E}[X_2]\big)\Big\}\leq \alpha\Big]\geq \beta^2$

I have tried doing it this way.

$P\Big[\max \Big\{f\big(\mathbb{E}[X_1]\big),f\big(\mathbb{E}[X_2]\big)\Big\}\leq \alpha\Big]=P\Big[f\big(\mathbb{E}[X_1]\big)\leq \alpha, f\big(\mathbb{E}[X_2]\big)\leq \alpha\Big]$

Now, because $X_1$ and $X_2$ are independent,
$P\Big[f\big(\mathbb{E}[X_1]\big)\leq \alpha, f\big(\mathbb{E}[X_2]\big)\leq \alpha\Big]=P\Big[f\big(\mathbb{E}[X_1]\big)\leq \alpha\Big]P\Big[f\big(\mathbb{E}[X_2]\big)\leq \alpha\Big]\geq \beta^2$

Is my approach correct?

Answer 1

Yes your approach is correct.

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