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4. Standard Deviation
– sd(X) =
X
Properties of Standard Deviation:
sd(c)=0
sd(aX+b) = a sd(x)
Measures of Distribution Shape:
5. Skewness 
measure of lack of symmetry
=
E
[
X
−
X
3
]
X
3
when skewness=0 the distribution is symmetric
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View Full Document 6. Kurtosis
thickness of the distribution tails
=
E
[
X
−
X
4
]
X
4
Measures of Association – For two or more r.v.s:
7. Covariance
–measures the amount of linear
dependence between two r.v.s
Cov
X,Y
=
E
[
X
−
X
]=
XY
=
E
[
XY
]−
X
Y
if Cov(X,Y)>0 then X and Y move in the same
direction.
Properties of Covariance:
When X and Y are independent, Cov(X,Y) = 0
Cov(aX+b, cY+d) =
ac Cov(X,Y)

XY

≤
X
Y
8. Correlation Coefficient
direction of relationship between two r.v.s
Corr
X,Y
=
Cov
X,Y
sx
X
sd
Y
=
XY
X
Y
=
XY
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View Full Document Properties of Correlation Coefficient:
1≤Corr(X,Y)≤1
Corr(X,Y)=0 implies X & Y are uncorrelated
Corr(X,Y)=1 implies perfect positive linear
relationship
Corr(aX+b, cY+b) = Corr(X,Y) if ac>0
properties.
9. Conditional Expectation
=conditional mean
E
[
YX
=
x
]=
∑
i
=
1
n
y
i
f
YX
y
i
x
Properties of Conditional Expectation:
E[g(X)X] = g(X) for any function g(.)
E[g(X)Y+h(X)X] = g(X)E[YX] + h(X)
E[YX] = E[Y]
if X and Y are independent
Law of Iterated Expectations
 if we take the
expectation of the mean of Y conditional on X,
across all x
i
, then we will simply have the
unconditional mean of Y
E
[
E
[
YX
]]=
∑
i
=
1
n
E
[
YX
=
x
i
]
P
X
=
x
i
=
E
[
Y
]
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View Full Document 10. Conditional Variance
– variance of the
conditional distribution of one r.v. Given
another
Var
YX
=
x
=
E
[
Y
2
X
=
x
]−
E
[
YX
=
x
]
2
=
∑
i
=
1
n
y
i
−
E
[
YX
=
x
]
2
f
YX
y
i
x
if X & Y are independent, then Var(YX) = Var(Y)
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This note was uploaded on 01/26/2012 for the course ECON 401 taught by Professor Burbidge,john during the Fall '08 term at Waterloo.
 Fall '08
 Burbidge,John

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