Homework 4

In all the problems f will denote a density, F the distribution function and c the constant needed to make the function a density.

Problem 1

Let (X,Y) be a random vector with joint density \(f(x,y) = cx/y^2, 1<x<y<2\).

1. Find c

2. Find F_X(1.5)

3. Find \(P(Y<1.5|X=1.25)\)

Problem 2

We roll three fair dice. Let X be the smallest of the three rolls and Y the largest. Find the joint density of (X,Y). Find F(2,4).

Problem 3

Say \(X\sim U[0,1]\) and \(Y|X=x\sim U[x, 1]\)

1. Find \(f_{X,Y}(x,y)\)

2. Find \(f_{Y}(y)\)

3. Find \(f_{X|Y=y}(x|y)\) and \(F_{X|Y=0.6}(0.4|0.6)\)