The size of the radius of a circle is uniform on the interval $[a,b]$ $(0<a<b)$ find the distribution for the probabilities of the circles area, it's mathematical expectation and variance.
I have no clue on how to start this problem.
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1It will be similar to this question. – angryavian May 09 '17 at 20:32
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HINT Here, $R \sim U(a,b)$ and $A = \pi R^2$, so the cdf of $A$ is $$ F_A(x) = \mathbb{P}[A \le x] = \mathbb{P}\left[R \le \sqrt{x/\pi}\right] = F_R \left(\sqrt{x/\pi}\right) $$
Can you plug into this the value of $F_R$ and use the fact that $f_A(x) = F_A'(x)$ to derive the pdf, and then the expectation and variance?
gt6989b
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1You could also use: $\displaystyle\mathsf E( g(R)) ~=~ \int_a^b \frac {g(r)}{b-a}\operatorname d r$ – Graham Kemp May 10 '17 at 00:24
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@GrahamKemp if this were an answer, I would upvote that and recommend instead of mine -- much easier to take 2 quick integrals than bother with the $\sqrt{\cdot}$ in my answer... – gt6989b May 10 '17 at 14:55