Cdf of minimum of two iid random variables

Question

I am struggling with the following sentence:

Using the fact that the cumulative distribution of the minimum of two i.i.d. random variables can be expressed as $1 - (1 - F(x))^2$....

Can anyone show me why this is the case?

Answer 1

It's a common trick. If $X = \min(Y_1,Y_2)$ and $F$, $F_X$ are the CDFs of the $Y_i$s and $X$, respectively, then \begin{align*} F_X(x) &= 1 - P(X > x) \\ &= 1- P(Y_1 > x, Y_2 > x) \\ &= 1 - P(Y_1 > x)P(Y_2 > x) \text{ independence}\\ &= 1 - [1-F(x)][1-F(x)] \text{ identicalness}. \end{align*}