Let a, b, c, d be any numbers with a<b and c<d. Let k be a constant, and let X and Y be jointly continuous with joint probability density function

In other words, f(x, y) is constant on the rectangle a<x<b and c<y<d, and zero off the rectangle.

a. Show that

b. Show that the marginal density of X is fX(x) = 1/(b - a) for a<x<b.

c. Show that the marginal density of Y is fY(y) = 1/(d - c) for c<y<d.

d. Use parts (a), (b); and (c) to show that X and Y are independent.

Answer

Step 1 of 5</p>

Here given the joint probability function with constant K

From that we need to find the value of k

Marginal probabilities functions of X and Y

And we have to show that X and Y independent

The given function is

=0 , otherwise

Step 2 of 5</p>

a) We know that total probability is one

Step 3 of 5</p>

b) the marginal density function of X is

=

=

=1/(b-a)

Hence 1/(b-a) for a<x<b