# Cheat Sheet For Probability Theory

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Ingmar Land, October 8, 2005
Cheat Sheet for Probability Theory
Ingmar Land
1
Scalar-valued Random Variables
Consider two real-valued random variables (RV) X and Y with the individual probabil-
ity distributions p
(x) and p
(y), and the joint distribution p
(x, y). The probability
X
Y
X,Y
distributions are probability mass functions (pmf) if the random variables take discrete
values, and they are probability density functions (ptf) if the random variables are con-
tinuous. Some authors use f () instead of p(), especially for continuous RVs.
In the following, the RVs are assumed to be continuous. (For discrete RVs, the integrals
have simply to be replaced by sums.)
• Marginal distributions:
(x) =
(x, y) d y
(y) =
(x, y) d x
p
p
p
p
X
X,Y
Y
X,Y
• Conditional distributions:
(x, y)
(x, y)
p
p
X,Y
X,Y
(x|y) =
(y|x) =
p
p
X|Y
Y |X
(y)
(x)
p
p
Y
X
for p
(x) = 0 and p
(y) = 0
X
Y
• Bayes’ rule:
(x, y)
(x, y)
p
p
X,Y
X,Y
(x|y) =
(y|x) =
p
p
X|Y
Y |X
(x , y) d x
(x , y) d y
p
p
X,Y
X,Y
• Expected values (expectations):
E g
(X) :=
(x) p
(x) d x
g
X
1
1
E g
(Y ) :=
(y) p
(y) d y
g
Y
2
2
E g
(X, Y ) :=
(x, y) p
(x, y) d x d y
g
X,Y
3
3
for any functions g
(.), g
(.), g
(., .)
1
2
3