Question 6 - Week 5 Practice

Passage

Let

(X,Y)\sim \text{Uniform}(R)

, where

R=\{(x,y):x^2+y^2<9,\ x>0,\ y>0\}

. What is the marginal density function

f_X(x)

? Solution outline with blanks: The joint distribution of

X

and

Y

is defined as

f_{XY}(x,y)=1/A

, where

A

represents __A__ and its value is __B__. The marginal density function

f_X(x)

can be calculated by integrating over

y

from

0

\sqrt{9-x^2}

, using __D__. This gives

f_X(x)=\dfrac{\text{__E__}\sqrt{\text{__C__}-x^2}}{9\pi}

. Consider the following options for the blanks: (1)

9\pi/4

(2)

4/(9\pi)

(3) Area of

R

(4)

f_{XY}(x,y)\,dx

(5)

1/\text{Area of }R

(6)

(\text{Area of }R)^2

(7)

f_{XY}(x,y)\,dy

(8)

f_X(x)\,dy

(9)

9

(10)

4

(11)

3

(12)

4\pi

Question

Enter the correct option number for __D__.

Passage question