Homework 14: Independent Random Variables

1. Section 3.5, #7, 11.
2. Let \(X\) and \(Y\) be jointly continuous with joint pdf \(f_{X,Y}(x,y)=(24/5)(x+y)\), for \(0\leq 2y\leq x\leq 1\).
   1. Find \(f_X(x)\) and \(f_Y(y)\).
   2. Are \(X\) and \(Y\) independent?
3. A point is chosen at random from the interior of a circle whose equation is \(x^2+y^2\leq 4\). Let the random variables \(X\) and \(Y\) denote the \(x\)- and \(y\)-coordinates of a sampled point.
   1. Find \(f_{X,Y}(x,y)\).
   2. Find \(f_X(x)\) and \(f_Y(y)\).
   3. Are \(X\) and \(Y\) independent?
4. A hat contains 3 red and 6 white slips of paper. You draw 2 slips at random, without replacement. Let \(X\) equal 1 if the first slip is red, and \(X\) equal 0 otherwise. Let \(Y\) equal 1 if the second slip is red, and \(Y\) equal 0 otherwise. Use the definition of independence of random variables to determine whether \(X\) and \(Y\) are independent.