Bivariate or joint probability distributions

10
(ii) Var( aX + bY) = a Var( X) + b Var( Y) + 2ab cov( X, Y).
Result (i) can be extended to any n random variables X 1
, X 2
,......., X n
E a X + a X + ....... + a X = a E X + a E X + ........ + a E X
( ) ( ) ( ) ( )
1 1 2 2 n n 1 1 2 2
n n
When X and Y are independent, then
(iii) Var( aX + bY) = a 2 Var( X) + b 2 Var( Y)
= so cov( X, Y ) = 0
(iv) E( XY ) E( X) E( Y )
Results (iii) and (iv) can be extended to any n independent random variables
X 1
, X 2
,......., X n
(iii)* Var( a X a X ....... a X )
+ + + =
1 1 2 2
n
n
2
2
( ) + ( ) + ........ + ( )
2
a Var X a Var X a Var X
1
1 2
(iv)* E( X X ..... X ) = E( X ). E( X )........
E( X )
1 2 n
1 2
n
2
n
n