Probability | Part IA, 2001

A random point is distributed uniformly in a unit circle D\mathcal{D} so that the probability that it falls within a subset AD\mathcal{A} \subseteq \mathcal{D} is proportional to the area of A\mathcal{A}. Let RR denote the distance between the point and the centre of the circle. Find the distribution function FR(x)=P(R<x)F_{R}(x)=P(R<x), the expected value ERE R and the variance VarR=ER2(ER)2\operatorname{Var} R=E R^{2}-(E R)^{2}.

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

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

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

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