A circle with a random radius R∼Unif(0,1) is generated. Let A be its area.
(a) Find the mean and variance of A, without first finding the CDF or PDF of A.
(b) Find the CDF and PDF of A.

So, quite obviously, we know the equation of a circle as A = ${\pi}r^2$. If we are given a uniform distribution, can we not just find $E(R)$ and then, using that value, find $E(A)$, and likewise for the variance? And, if we have $F(A)$, is this the CDF or PDF?


Some guide:

  • Notice that $$\mathbb{E}(A) \neq \pi (\mathbb{E}(R))^2$$ in general. Hence we can't say that since $\mathbb{E}(R)= \frac12$, hence $\mathbb{E}(A)=\frac{\pi}{2^2}$, this is not the right approach.

  • We have, $$\mathbb{E}(A) = \pi\mathbb{E}(R^2)$$ and to evaluate $\mathbb{E}(R^2)$, you might like use for the formula of $Var(X)=\mathbb{E}(X^2)-\mathbb{E}(X)^2$.

  • To find variance of $A$, you might want to use the same formula as well.

  • $F(A)$ usually denote CDF.
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