Expected Values
Def. Let X and Y be jointly distributed random variables with pmf p(x,y) or pdf f(x,y) according to whether the variables are discrete or continuous. Then the expected value of a function h(X,Y), denoted by E [h(X,Y)] or  is given by:
E [h(x,y)] =  if X and Y are discrete
=  if X and Y are continuous
|
Ex. 11 The joint density for the random variable (X,Y) of Example 1 is given in table below. X denotes the number of defective welds and Y the number of improperly tightened bolts produced per car by assembly line robots.
Compute E [X], E [Y], E [X +Y], and E [XY]
-
Y
X
|
0 1 2 3
|
|
0
1
2
|
0.840 0.030 0.020 0.010
0.060 0.010 0.008 0.002
0.010 0.005 0.004 0.001
|
0.900
0.080
0.020
|
|
0.910 0.045 0.032 0.013
|
1.000
|
Ex.12 The joint density for the random variable (X,Y) where X denotes the calcium level and Y the cholesterol level in the blood of a healthy individual is given by
f (x,y) = 1 / 240, 8.5 x 10.5
y 240
Find E [X], E [Y], E [XY] (x, y are in units of mg/dl)
Covariance - direction of the association
When two random variables X and Y are not independent, it is frequently of interest to measure how strongly they are related to one another., in what direction.
Def. The covariance between two random variables X and Y is:
Cov (x,y) = E [(x - x )(Y - y )]
=  -------- X,Y discrete
=  -----------X,Y continuous
If X and Y have a strong positive relationship, meaning that X Y and X Y, then (x-x) (y-y) will be positive. Thus Cov (X, Y) should be positive.
If X and Y have a negative relationship, the signs of (x-x) (y-y) will tend to be opposite. Thus Cov (X, Y) should be negative.
If X and Y are not strongly related, positive and negative products will tend to cancel one another, yielding a covariance near 0.
(a) positive covariance (b) negative covariance (c) covariance near zero
Share with your friends: |
