Can $\prod_{l = 1}^L \prod_{m=1}^M \text{P}_r(X\leq\gamma_{th})$ be written as $[F_X(\gamma_{th})]^{L\cdot M}$, where $F_X(\cdot)$ is CDF of $X$?

Question

I am dealing with big Pi notation as shown below:

\begin{align} P = \prod_{l = 1}^L \prod_{m=1}^M \text{P}_r(X\leq\gamma_{th}) \tag1 \end{align}

where $X$ is a random variable. And all others are constant.

My query is

Can we write eq (1) as shown by eq (2). \begin{align} P = [F_X(\gamma_{th})]^{L\cdot M} \tag2 \end{align} where $F_X(\cdot)$ is CDF of $X$?

Any help in this regard will be highly appreciated.

Answer 1

Yes, it's true. Put $a:=\text{P}_r(X\leq\gamma_{th})$. In $\prod_{m=1}^M a$ we have a product of $M$ the same factors $a$. Therefore $\prod_{m=1}^M a = a^M$. The same is with the outer product.