ADVERTISEMENT

1

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

ADVERTISEMENT

0 votes

Parent category: Education