Find density function of area

Question

I just started taking my first statistics course and I seem to be having trouble understanding this concept and answering this question.

If the radius of a circle is an exponential random variable, how do you find the density function of the area?

I am wondering if anyone can point me to any resources or help me understand this a little bit since I seem to be stuck currently.

Answer 1

Go via the cumulative distribution function, since this represents a probability, which the density function doesn't (at least not directly). We have $$P(A\,{\le}\,a)=P(\pi R^2\,{\le}\,a)=P(R\,{\le}\,\sqrt{a/\pi}) =\int_0^{\sqrt{a/\pi}} \lambda e^{-\lambda r}\,dr =1-e^{-\lambda\sqrt{a/\pi}}$$ for $a>0$; and then the density function for $A$ is the derivative of this with respect to $a$, which I'm sure you can do yourself.

Answer 2

So $X\sim\mathcal {Exp}(\lambda)$, then $f_X(x)= \lambda~e^{-\lambda x}\mathbf 1_{x\in[0;\infty)}$

Given $Y=\pi X^2$ then $X={+\sqrt{Y/\pi~}}$ (...and we have a bijection)

By the transformation theorem:$$f_Y(y) = f_X(\sqrt{y/\pi~})~\left\lvert\dfrac{\mathrm d~\sqrt{y/\pi~}}{\mathrm d~y\qquad\;}\right\rvert \\ = \dfrac{\lambda e^{-\lambda\sqrt{y/\pi~}}}{2\sqrt{\pi y~}}\mathbf 1_{y\in [0;1)}$$