Homework 4

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 $F_{X|Y=0.6}(0.4|0.6)$