Random variables x and y are independent with PDFs fXx13 ex3

Random variables x and y are independent with PDF\'s fX(x)={1/3 e^-x/3 x GE 0, 0 otherwise fY(y)={ 1/2e^-y/2 y GE 0, 0 otherwise Find P(x > y). Find E[xy]. What is Cos(x,y)?

Solution

(a)

integral (0->infinity) ( integral(y->infinity) (1/3 e^(-x/3))dx ) dy

=integral (0->infinity) ( integral(y->infinity) (1/3 * e^(-x/3) * (-3) ) dy

=integral (0->infinity) ( (y->infinity) ( -e^(-x/3) ) dy

=integral (0->infinity) (e^(-y/3) dy

=(0->infinity) (-3e^(-y/3)

= (0 + 3)

= 3 Answer

(b)

E(X) = integral (0->infinity) 1/3 e^(-x/3))dx

= (0->infinity) 1/3 * e^(-x/3) * -3

= (0->infinity) -e^(-x/3)

= 0 +1

=1

E(Y) = integral (0->infinity) 1/2 e^(-y/2))dx

= (0->infinity) 1/2 * e^(-y/2) * -2

= (0->infinity) -e^(-y/2)

= 0 +1

=1

Therefore,

E(XY) = E(X)*E(Y) = 1*1 =1 Answer

(c)

COV(x,y) = E(XY) - E(X)*E(Y)

= 1 - 1*1

=1-1

= 0 Answer