2.II.10F

A random point is distributed uniformly in a unit circle $\mathcal{D}$ so that the probability that it falls within a subset $\mathcal{A} \subseteq \mathcal{D}$ is proportional to the area of $\mathcal{A}$. Let $R$ denote the distance between the point and the centre of the circle. Find the distribution function $F_{R}(x)=P(R<x)$, the expected value $E R$ and the variance $\operatorname{Var} R=E R^{2}-(E R)^{2}$.

Let $\Theta$ be the angle formed by the radius through the random point and the horizontal line. Prove that $R$ and $\Theta$ are independent random variables.

Consider a coordinate system where the origin is placed at the centre of $\mathcal{D}$. Let $X$ and $Y$ denote the horizontal and vertical coordinates of the random point. Find the covariance $\operatorname{Cov}(X, Y)=E(X Y)-E X E Y$ and determine whether $X$ and $Y$ are independent.

Calculate the sum of expected values $E \frac{X}{R}+i E \frac{Y}{R}$. Show that it can be written as the expected value $E e^{i \xi}$ and determine the random variable $\xi$.

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