Question

Let $(X, Y)$ be uniformly distributed in a circle of radius $r$ about the origin. That is, their joint density is given by $$ f(x, y)=\frac{1}{n r^{2}}, \quad 0 \leq x^{2}+y^{2} \leq r^{2} $$ Let $R=\sqrt{X^{2}+Y^{2}}$ and $\theta=\arctan Y / X$ denote their polar coordinates. Show that $R$ and $\theta$ are independent with $\theta$ being uniform on $(0,2 \pi)$ and $P[R<a\}=a^{2} / r^{2}, 0<a<r$

          Let $(X, Y)$ be uniformly distributed in a circle of radius $r$ about the origin. That is, their joint density is given by
$$
f(x, y)=\frac{1}{n r^{2}}, \quad 0 \leq x^{2}+y^{2} \leq r^{2}
$$
Let $R=\sqrt{X^{2}+Y^{2}}$ and $\theta=\arctan Y / X$ denote their polar coordinates. Show that $R$ and $\theta$ are independent with $\theta$ being uniform on $(0,2 \pi)$ and $P[R<a\}=a^{2} / r^{2}, 0<a<r$

Added by William L.

Probability with Applications in Engineering, Science, and Technology

Matthew A. Carlton • Jay L. Devore 2nd Edition

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Solved by Expert Ameer Said

12/09/2021

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Let $(X, Y)$ be uniformly distributed in a circle of radius $r$ about the origin. That is, their joint density is given by $$ f(x, y)=\frac{1}{n r^{2}}, \quad 0 \leq x^{2}+y^{2} \leq r^{2} $$ Let $R=\sqrt{X^{2}+Y^{2}}$ and $\theta=\arctan Y / X$ denote their polar coordinates. Show that $R$ and $\theta$ are independent with $\theta$ being uniform on $(0,2 \pi)$ and $P[R<a\}=a^{2} / r^{2}, 0<a<r$