Let $X$ be a random variable with p.d.f. $$f_X(x) = \frac{1}{\beta}e^{-x/\beta} 1_{[0,∞)}(x)$$
, where $\beta > 0$.
For a positive number $b$, let us define a random variable $Y$ as
$$Y =\begin{cases} 1 &,\text{ if } X > b \\ 0 &,\text{ otherwise }\end{cases}$$
For what value of $b$ will $Y$ have a Bernoulli$(1/2)$ distribution?
I'm other words, what is the median of $X$? Why, it's $\beta\ln 2$, of course.