Expectations
Expectations
Example (Problem 75 revisit)
A restaurant serves three fixed-price dinners costing $12, $15, and
$20. For a randomly selected couple dinning at this restaurant, let
X = the cost of the man’s dinner and Y = the cost of
the woman’s dinner. If the joint pmf of X and Y is assumed to
be
y
p(x, y )
12 15 20
12 .05 .05 .10
x
15 .05 .10 .35
20 0 .20 .10
What is the expected total expense for that couple?
Expectations
Example (Problem 75 revisit)
A restaurant serves three fixed-price dinners costing $12, $15, and
$20. For a randomly selected couple dinning at this restaurant, let
X = the cost of the man’s dinner and Y = the cost of
the woman’s dinner. If the joint pmf of X and Y is assumed to
be
y
p(x, y )
12 15 20
12 .05 .05 .10
x
15 .05 .10 .35
20 0 .20 .10
What is the expected total expense for that couple?
Let Z = the total expense for that couple. Then
Z = X +Y.
Expectations
Expectations
p(x, y )
x
12
15
20
12
.05
.05
0
y
15
.05
.10
.20
20
.10
.35
.10
Z =X +Y
Expectations
p(x, y )
x
12
15
20
12
.05
.05
0
y
15
.05
.10
.20
20
.10
.35
.10
Z =X +Y
E (Z ) = .05(12 + 12) + .05(12 + 15) + .10(12 + 20)
+ .05(15 + 12) + .10(15 + 15) + .35(15 + 20)
+ 0(20 + 12) + .20(20 + 15) + .10(20 + 20)
= 33.35
Expectations
Expectations
Definition
Let X and Y be jointly distributed rv’s 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 µh(X ,Y ) , is given by
(P P
h(x, y ) · p(x, y )
E [h(X , Y )] = R ∞x R y∞
−∞ −∞ h(x, y ) · f (x, y )dxdy
if X and Y are discrete
if X and Y are continuo
Expectations
Expectations
Example (Problem 12)
Two components of a minicomputer have the following joint pdf
for their useful lifetimes X and Y :
(
xe −x(1+y ) x ≥ 0 and y ≥ 0
f (x, y ) =
0
otherwise
Expectations
Example (Problem 12)
Two components of a minicomputer have the following joint pdf
for their useful lifetimes X and Y :
(
xe −x(1+y ) x ≥ 0 and y ≥ 0
f (x, y ) =
0
otherwise
If the lifetime of the minicomputer is the sum of the lifetimes of
the two components, then what is the expected lifetime of the
minicomputer?
Covariance
Covariance
Covariance
Covariance
Covariance
Covariance
Covariance
Covariance
Definition
The covariance between two rv’s X and Y is
Cov (X , Y ) = E [(X − µX )(Y − µY )]
(P P
y (x − µX )(y − µY )p(x, y )
= R ∞x R ∞
−∞ −∞ (x − µX )(y − µY )f (x, y )dxdy
X , Y discrete
X , Y continuou
Covariance
Definition
The covariance between two rv’s X and Y is
Cov (X , Y ) = E [(X − µX )(Y − µY )]
(P P
y (x − µX )(y − µY )p(x, y )
= R ∞x R ∞
−∞ −∞ (x − µX )(y − µY )f (x, y )dxdy
X , Y discrete
X , Y continuou
Remark: The covariance depends on both the set of possible pairs
and the probabilities.
Covariance
Definition
The covariance between two rv’s X and Y is
Cov (X , Y ) = E [(X − µX )(Y − µY )]
(P P
y (x − µX )(y − µY )p(x, y )
= R ∞x R ∞
−∞ −∞ (x − µX )(y − µY )f (x, y )dxdy
X , Y discrete
X , Y continuou
Remark: The covariance depends on both the set of possible pairs
and the probabilities.
Proposition
Cov (X , Y ) = E (XY ) − µX · µY
Covariance
Covariance
Example (Problem 75 revisit)
A restaurant serves three fixed-price dinners costing $12, $15, and
$20. For a randomly selected couple dinning at this restaurant, let
X = the cost of the man’s dinner and Y = the cost of
the woman’s dinner. If the joint pmf of X and Y is assumed to
be
y
p(x, y )
12 15 20
12 .05 .05 .10
x
15 .05 .10 .35
20 0 .20 .10
Covariance
Example (Problem 75 revisit)
A restaurant serves three fixed-price dinners costing $12, $15, and
$20. For a randomly selected couple dinning at this restaurant, let
X = the cost of the man’s dinner and Y = the cost of
the woman’s dinner. If the joint pmf of X and Y is assumed to
be
y
p(x, y )
12 15 20
12 .05 .05 .10
x
15 .05 .10 .35
20 0 .20 .10
Cov (X , Y ) = E (XY ) − µX · µY = 276.7 − 15.9 · 17.45 = −0.755
Covariance
Covariance
If we change the unit for the previous example from dollar to cent,
then the joint pmf would be
y
1200 1500 2000
p(x, y )
1200 .05
.05
.10
.10
.35
x
1500 .05
0
.20
.10
2000
Covariance
If we change the unit for the previous example from dollar to cent,
then the joint pmf would be
y
1200 1500 2000
p(x, y )
1200 .05
.05
.10
.10
.35
x
1500 .05
0
.20
.10
2000
And correspondingly,
Cov (X , Y ) = E (XY ) − µX · µY = 7550
Covariance
Covariance
Definition
The correlation coefficient of X and Y , denoted by Corr (X , Y ),
ρX ,Y or just ρ is defined by
ρX ,Y =
Cov (X , Y )
σX · σY
Covariance
Definition
The correlation coefficient of X and Y , denoted by Corr (X , Y ),
ρX ,Y or just ρ is defined by
ρX ,Y =
Cov (X , Y )
σX · σY
e.g. for the previous example, the correlation coefficient of X and
Y is
−0.755
ρ=
= −0.09
2.91 · 2.94
Covariance
Covariance
Proposition
1. Corr (aX + b, cY + d) = Corr (X , Y ) if a · c > 0.
2. −1 ≤ Corr (X , Y ) ≤ 1.
3. ρ = 1 or −1 iff Y = aX + b for some a and b with a 6= 0.
4. If X and Y are independent, then ρ = 0. However, ρ = 0
does not imply that X and Y are independent