Given the joint PDF of X and Y fxy x y 61 x y 0 SolutionG

Given the joint PDF of X and Y f_xy (x, y) = {6(1 - x - y), 0

Solution

Given that ,

f(x.y) = 6 ( 1 - x - y ) 0 x 1 and  0 y 1

Now first we have to find E(XY).

E(XY) = xy * f(x,y) dx dy

= xy 6 ( 1 - x - y ) dx dy

= x dx   6y (1 - x - y) dy

= x dx   [ 6y - 6xy - 6y² ]  dy

=   x dx [ 6*[y²/2] - 6x * [y² / 2] - 6 * [y³ /3] (y is from 0 to 1)

E(XY) =   x dx [ 3y² - 3xy² - 2y³ ]

=   x dx [ 3(1)² - 3x(1)² - 2(1)³ ]

=   x dx [3 - 3x - 2]

E(XY) =   x dx [1 - 3x]

=  ( x - 3x² ) dx

= [ x² / 2 - 3(x³ / 3) ] (x is from 0 to 1)

E(XY) = [ 1² / 2 - 3(1³ / 3) ]

= 1/2 - 1

E(XY) = -1/2

Now we hae to find Cov(X,Y)

Cov(X,Y) = E(XY) - E(X) * E(Y)

Now we calculat firstmarginal distribution of X and Y.

Marginal distribution of X is,

f(x) =   f(x,y) dy (y is from 0 to 1)

= 6(1-x-y) dy

=   (6 - 6x - 6y) dy

=  [ 6y - 6xy - 6 (y² /2) ] (y is from 0 to 1)

= [ 6(1) - 6x(1) - 6/2 (1²) ]

= 6 - 6x - 3

f(x) = 3 - 6x x is from 0 to 1

Similarly we find marginal distribution of Y:

f(y) = f(x,y) dx (x is from 0 to 1)

= 6(1-x-y) dx

=   (6 - 6x - 6y) dx

=  [ 6x - 6 (x² /2) - 6xy] (x is from 0 to 1)

= [ 6(1) - 6/2 (1²) - 6y(1) ]

= 6 - 6y - 3

f(x) = 3 - 6y y is from 0 to 1

Now we find E(X) and E(Y).

E(X) = x * f(x) dx

= x * (3- 6x) x is from 0 to 1

E(X) = 3x - 6x²   dx

= 3 [ x² / 2 ] - 6 [x³ / 3]

= 3 [ 1/2 ] - 6 [ 1/3 ]

= 3/2 - 2

E(X) = -1/2

E(Y) = y * f(y) dy

= y * (3- 6y) y is from 0 to 1

E(Y) = 3y - 6y²   dy ( yis from 0 to 1)

= 3 [ y² / 2 ] - 6 [y³ / 3]

= 3 [ 1/2 ] - 6 [ 1/3 ]

= 3/2 - 2

E(X) = -1/2

Cov(X,Y) = E(XY) - E(X)E(Y)

= -1/2 - (-1/2 * -1/2)

= -0.75

Now we have to find var (x) and var(y).

var(x) = E(x² ) - [ E(x) ]²

var(y) = E(y² ) - [ E(y) ]²

Now we have to find expected of x² and y².

E(x²) = x² * f(x) dx

= x² * (3- 6x) x is from 0 to 1

=   3x² - 6x³  dx

= [ 3*(x³/3) - 6*(x⁴ /4) ]

= 3 (1/3) - 6(1/4) = -1/2

E(y²) =   y² * f(y) dy

= y² * (3- 6y) y is from 0 to 1

=   3y² - 6y³  dy

= [ 3*(y³/3) - 6*(y⁴ /4) ]

= 3 (1/3) - 6(1/4) = -1/2

var(x) = -1/2 - (-1/2)² = 0.75

var(y) = 0.75

sd(x) = sqrt(var(x)) = sqrt(0.75) = 0.8660

sd(y) = 0.8660

_xy = Cov(X,Y) / sd(X) * sd(Y)

= -0.75 / 0.8660 * 0.8660

= -1