I am solving a question given as:
Let $X_1,X_2,\dots,X_N$ be i.i.d random variables. Let $Y = \sum_{i=1}^{N} X_i$ , Prove that:
$$\Pr(Y\geq a) \leq \min_{t>0} \left\{ e^{-ta}\prod_{i=1}^N \mathbb{E}\left[e^{tX_i}\right] \right\}. $$
I tried it in the following manner:
From Markov's inequality, we know that for any $t > 0$
$\Pr{(Y \geq a)} = \Pr(e^{tY} \geq e^{ta}) \leq \frac {\mathbb{E}[e^{tY}]}{e^{ta}}$
$=e^{-ta} \mathbb{E}[e^{tY}] = e^{-ta}\mathbb{E}[e^{t\sum_{i=1}^{N} X_i}]$
$=e^{-ta}\prod_{i=1}^N \mathbb{E}[e^{tX_i}]$
But I don't know how the minimum part came into the solution.
Any suggested reading or hint will be very helpful. Thanks in advance!