Cheat Sheet For Probability Theory Page 3

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3
Ingmar Land, October 8, 2005
• Conditional expectations:
E g(X)|Y = y :=
(x|y) d x
g(x)p
X|Y
Law of total expectation:
E E g(X)|Y
=
E g(X)|Y = y p
(y) d y
Y
=
(x|y) d xp
(y) d y
g(x)p
X|Y
Y
=
(x|y)p
(y) d y d x
g(x)
p
X|Y
Y
=
(x) d x
g(x)p
X
= E g(X)
• Special conditional expectations:
– Conditional mean:
:= E X|Y = y =
(x|y) d x
µ
xp
X|Y =y
X|Y
– Conditional variance:
2
2
Σ
:= E (X − µ
)
|Y = y =
(x − µ
)
(x|y) d x
p
XX|Y =y
X
X
X|Y
– Relation:
2
2
E X
|Y = y = Σ
+ µ
XX|Y =y
X|Y =y
• Sum of two random variables: Let Z := X + Y ; then
(z) ∗ p
(z) = p
(z)
p
Z
X
Y
where ∗ denotes convolution. The proof uses the characteristic functions.
• The RVs are called independent if
(x) · p
(x, y) = p
(y).
p
X,Y
X
Y
This condition is equivalently to the condition that
(X) · g
(X) · E g
E g
(Y ) = E g
(Y )
1
2
1
2
for all (!) functions g
(.) and g
(.).
1
2

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