Question

Suppose X and Y have joint probability density function f(x,y) = 6(x-y) when 0<y<x<1 and f(x,y) = 0 otherwise.

(a) Indicate with a sketch the sample space in the x-y plane

(b) Find the marginal density of X, fX(x)

(c) Show that fX(x) is properly normalized, i.e., that it integrates to 1 on the sample space of X

(d) Find the marginal density of Y, fY(y)

(e) Show that fY(y) is properly normalized, i.e., that it integrates to 1 on the sample space of Y

(f) Are X and Y independent? Why or why not?

Answer 1

a) 0<y<x<1, indicates that both x and y have values between 0 and 1 and y is less than x

We can draw the following sketch of the sample space in x-y plane. The shaded region is the sample space

b) The marginal density of X is

Formally, the marginal pdf of X is

c) We integrate the pdf of X over the support region

Since the density of X integrates to 1 on the sample space of X, we can say that is properly normalized

d) The marginal density of Y is

Formally, the marginal pdf of Y is

e) We integrate the pdf of Y over the support region

Since the density of Y integrates to 1 on the sample space of Y, we can say that is properly normalized

f) We can say that X and Y are independent if the product of marginal pdfs is equal to the joint pdf

The product of the marginal pds of X,Y is

The joint pdf of X,Y is

ans: Since the joint pdf is not equal to the product of marginal pdfs of X,Y, we can say that X and Y are not independent.