Problem 1

Say \((X,Y)\) is a discrete rv with joint pdf \(f_{X,Y}(x,y)=(1-p)^2p^x; x,y \in \{0,1,..\}, y \le x, 0<p<1\). Let \(U=I(|X-Y|\le 1)\). Find the density of U.

Problem 2

Let \(X\sim U[-1,1]\).

  1. say \(Y=1-X^2\). Find the density of \(Y\).

  2. say \(Y=1-\left(2X-\frac{X}{|X|}\right)^2\). Find the density of Y.

Problem 3

Say the random vector \((X,Y)\) has joint density \(f(x,y)=\frac1{2\pi}\exp\{-(x^2+y^2)/2\}\). Let \(D=\sqrt{X^2+Y^2}\). Find the density of D.

Hint: what is the density of \(X^2\)?

Problem 4

Let \(X,Y\) be independent rv’s with \(P(X=k)=P(Y=k)=\frac1N\), \(k=1,...,N; N\ge 2\). Let \(M=\min\{X,Y\}\). Find the density of M.