Q.6.19

Question

Show that f(x, y) = 1/x, 0 < y < x < 1, is a joint density function. Assuming that f is the joint density function of X, Y, find

(a) the marginal density of Y;

(b) the marginal density of X;

Step-by-Step Solution

Verified

Answer

The joint density of X and Y is given by f(X,Y) is not independent.

1Step 1: Introduction

The joint density of X and Y is not independent.

2Step 2: Given Information

$T h r e e p o i n t s X 1, X 2, X 3 a r e s e l e c t e d a t r a n d o m o n a l i n e L F r o m t h e i n f o r m a t i o n, o b s e r v e t h a t t h e j o i n t d e n s i t y f u n c t i o n o f X a n d Y i s a s f o l l o w s : f (x, y) = \{\begin{cases} x e^{- (x + y)} & x > 0, y > 0 \\ 0 & O t h e r v i s e \end{cases} C h e c k w h e t h e r X a n d Y a r e i n d e p e n d e n t o r n o t . T h e m a r g i n a l d e n s i t y o f X i s, f_{x} (x) = \int_{0}^{\infty} f (x, y) d y \int_{0}^{\infty} x e^{- (x + y)} d y \int_{0}^{\infty} x e^{- x} e^{- y} d y x e^{- x} {(- e^{- y})}_{0}^{\infty} x e^{- x} (e^{- 0} - e^{- 0}) x e^{- x} C a l c u l a t e t h e m a r g i n a l d e n s i t y o f Y f_{r} (y) = \int_{0}^{\infty} f (x, y) d x = \int_{0}^{\infty} x e^{- (x + y)} d x = \int_{0}^{\infty} x e^{- x} e^{- y} d x = e^{- y} \int_{0}^{\infty} x e^{- x} d x = e^{- y} [- x {(e^{- x})}_{0}^{\infty} + \int_{0}^{\infty} e^{- x} d x] (since integration by parts) = \int_{0}^{\infty} x e^{- (x + y)} d x = e^{- y} [- x (e^{- x} - e^{- 0}) - {(e^{- x})}_{0}^{\infty}] = e^{- y} [1] = e^{- y} T h e r e f o r e, f_{x} (x) f_{Y} (y) = x e^{- x} e^{- y} = x e^{- x - y} = x e^{- (x + y)} = f (x, y) H e n c e, X a n d Y a r e i n d e p e n d e n t . N o w T h e s e r a n d o m v a r i a b l e s a r e n o t i n d e p e n d e n t . I f t h e y w e r e i n d e p e n d e n t, t h e i r j o i n t P D F w o u l d f a c t o r i z e f (x, y) = f (x) f (y) B u t f o r e v e r y p o i n t (x, y) \in (0, 1)^{2} s u c h t h a t y < x e w o u l d h a v e f_{x} (x) > 0 a n d f_{r} (y) > 0 o n t h e o t h e r h a n d f (x, y) = 0 T h a t l e a d s t o t h e c o n t r a d i c t i o n . H e n c e, t h e y a r e n o t i n d e p e n d e n t .$

Q.6.18

Q. 6.24

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