Query about probability expression containing max operator

Question

I came across an expression in one of the research paper.

$P_0 =$ $\text{Pr}\Bigl(\text{max}(\gamma_1,\gamma_2,...,\gamma_L)<\gamma_{th}\Bigr)$----(1)

$P_0 =\Bigl[F_{\gamma_l(\gamma_{th})}\Bigr]^L$ ----(2)

where $\text{Pr}$ denotes probability, $F$ denotes the CDF, $\gamma_l$ are random variables.

I am not getting how $L$ comes in power and $\text{max}$ gets removed in equation (2).

Any help in this regard will be highly appreciated.

Answer 1

Let $\{X_i\}_{i=1}^{n}$ be i.i.d. We have $$ P(\max(X_1, \dots, X_n) < t) = P(X_1 < t, X_2 < t, \dots X_n < t) = P(X_1<t)^n.$$