Here's an interesting question:
The area of a circle is exponentially distributed with parameter $\lambda$. Find the distribution of PDF of the radius of the circle. Then assume that the radius is exponentially distributed (with the same rate $\lambda$) and find the PDF of the circle’s area.
So an exponential distribution means that the density function would be along the lines of:
PDF(x) = f(x) = $\lambda e^{-\lambda x}dx$
then I think one would do something along these lines:
$P${$R<=r$} = $ \frac{1}{2\pi} \int_{B(0)}\int r^\frac{r^2}{2}dr$ = $\int^0_r r^{\frac{r^2}{2}} dr$
to find the PDF of the radius, right?
How would I apply this to find the PDF of the circle's area?