Cheat Sheet For Probability Theory Page 2
ADVERTISEMENT
1
2
3
4
5
62
Ingmar Land, October 8, 2005
• Some special expected values:
– Means (mean values):
:= E X =
(x) d x
:= E Y =
(y) d y
µ
x p
µ
y p
X
X
Y
Y
– Variances:
2
2
2
≡ Σ
:= E (X − µ
(x − µ
)
=
)
(x) d x
σ
p
XX
X
X
X
X
2
2
2
≡ Σ
:= E (Y − µ
(y − µ
)
=
)
(y) d y
σ
p
Y Y
Y
Y
Y
Y
Remark: The variance measures the “width” of a distribution. A small variance
means that most of the probability mass is concentrated around the mean value.
– Covariance:
≡ Σ
:= E (X − µ
)(Y − µ
)
σ
XY
XY
X
Y
(x − µ
)(y − µ
=
) p
(x, y) d x d y
X
Y
X,Y
Remark: The covariance measures how “related” two RVs are. Two indepen-
dent RVs have covariance zero.
– Correlation coefficient:
σ
XY
:=
ρ
XY
σ
σ
X
Y
– Relations:
2
2
E X
= Σ
+ µ
XX
X
2
2
E Y
= Σ
+ µ
Y Y
Y
E X · Y = Σ
+ µ
· µ
XY
X
Y
– Proof of last relation:
E XY = E ((X − µ
) + µ
)((Y − µ
) + µ
)
X
X
Y
Y
= E (X − µ
)(Y − µ
) − E (X − µ
)µ
− E µ
(Y − µ
) +
X
Y
X
Y
X
Y
+ E µ
µ
X
Y
− (E[X] − µ
− (E[Y ] − µ
= Σ
)µ
)µ
+ µ
µ
XY
X
Y
Y
X
X
Y
= Σ
+ µ
µ
XY
X
Y
This method of proof is typical.
ADVERTISEMENT
0 votes
Related Articles
Related forms
Related Categories
Parent category: Education