Bivariate or joint probability distributions

3.13 Expectations and variances
8
Discrete random variables
r
r
r
r
( ) = ∑ ∑ ( , ) = ∑ x ∑ p( x , y)
= x p
X ( x)
E X x p x y
x
y
x
r
r
r
r
( ) = ∑ ∑ ( , ) = ∑ y ∑ p( x , y)
= y pY
( y)
E Y y p x y
Examples
x
y
y
y
x
∑ r =1,2.....
x
∑ r =1,2....
y
2 2
Hence Var(X) = E( X ) − ( E( X ))
, Var(Y) = E( Y ) ( E( Y
)
Continuous random variables
∞
∞
r r r
( ) ∫ ∫ ( ) ∫ X ( )
− etc.
2 2
E X = x f x , y dx dy = x f x dx r = 1, 2 .....
−∞ −∞
∞
∞
r r r
( ) ∫ ∫ ( ) ∫ Y ( )
E Y = y f x , y dx dy = y f y dy r = 1, 2 .....
Examples
−∞ −∞
∞
−∞
∞
−∞
3.14 Expectation of a function of the r.v.'s X and Y
Continuous X and Y
Discrete X and Y
∞
∞
∫ ∫
E[ g( X, Y)] = g( x, y) f ( x, y)
dxdy
−∞ −∞
∞
∞
e . g . ⎡
E X ⎤ x
⎣
⎢ Y ⎦
⎥ = ∫ ∫
y f ( x, y)
dxdy
E[XY] =
−∞ −∞
∞
∞
∫ ∫
−∞ −∞
xy f ( x , y ) dxdy .
3.15 Covariance and correlation
Covariance of X and Y is defined as follows : Cov (X,Y) = σ XY
= E(XY) - E(X)E(Y).
Notes
(a) If the random variables increase together or decrease together, then the covariance
will be positive, whereas if one random variable increases and the other variable
decreases and vice-versa, then the covariance will be negative.
(b) If X and Y are independent r.v's, then E(XY) = E(X)E(Y) so cov(X, Y) = 0.
However
if cov(X,Y) = 0, it does not follow that X and Y are independent unless X and Y are