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If a point is picked uniformly at random inside a disc of radius $9$, find the CDF and density of the distance $T$ from the center for $0 < t < 9$ .

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1 Answer 1

We find the cumulative distribution function $F_T(t)$ of $T$. This is the probability that $T\le t$.

Now $T\le t$ if and only if the point picked is inside or on the circle with centre the origin and radius $t$. This circle has area $\pi t^2$. The whole circle has area $\pi(9^2)$. Since our distribution is uniform, we have $$F_T(t)=\Pr(T\le t)=\frac{\pi t^2}{\pi(9^2)}=\frac{t^2}{9^2}$$ when $0\lt t\lt 9$. For completeness, we should add that $F_T(t)=0$ if $t\le 0$, and $F_T(t)=1$ when $t\ge 9$, though this is not asked for.

For the density function $f_T(t)$, differentiate $F_T(t)$ with respect to $t$.

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