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Answered Same Day Dec 26, 2021

Solution

Robert answered on Dec 26 2021

118 Votes

Solution:
a) The value of joint PDF inside the inside the triangle:
( )

( ) {

Other way, we can write:
( )
) Calculating the marginal PDF of X and Y:
( )
{

∫ ( )

∫ ( )

( ) {

( ) ∫ ( )
( )
( ) ( )
Now:
( ) ( ) ( )
Therefore, X and Y are not independent, they are dependent.
Checking the co
elation between the two random variables:
[ | ] ∫ | ( | )
∫
( )
( )
[ | ] ∫

( )
[ | ]
( )

Similarly,
[ | ]
{

∫

∫

[ | ] {

( )

Using the total expectation theorem:
[ ] ∫ [ | ] ( )
∫( )( )
[ ]
Similarly,
[ ] ∫ [ | ] ( )
∫

∫
( )

( )

[ ]

Now:
[ | ] [ | ] {

( )

[ ] ∫ [ | ] ( )
∫

∫
( )

( )

[ ]

Other way of doing this is:
[ | ] [ | ]
( )

[ ] ∫ [ | ] ( )
∫ ( )( )
[ ]

The covariance can be calculated as:
( ) [ ] [ ] [ ]

( )

Since, the covariance is not...

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