TITLE: what is the median of the CDF with the form $F(x) = 1 - e^{-(x/3)^2}$, for $x \gt 0$
QUESTION [5 upvotes]: Consider The distribution of lifetimes, X(in months), of a particular type of component. The Cumulative distribution function (CDF) has the form
$$F(x)=\begin{cases}
1 - e^{-(x/3)^2},& \text{if} ~ x \gt 0\\\\
0,&\text{otherwise}.\\\\
\end{cases}$$
How to calculate the median?

REPLY [2 votes]: $F(x_0) = 1 - e^{-(x_0/3)^2}=0.5\implies -x_0^2/9=ln(0.5)\implies x_0=3 \sqrt{\ln(2)}$