Introduction; Review of Random Variables
Probability :: Set Theory :: Definition :: Conditional :: Random :: Distribution :: R.V.S.
Pure types of r.v.s
 Discrete r.v.s  an r.v. whose possible values can be enumerated.
 Continuous r.v.s  an r.v. whose distribution function can be written as the (regular) integral of another function
 Singular, but not discrete  Any other r.v.
Discrete r.v.s
A random variable that can take on at most a
countable
number of
possible values is said to be a discrete r.v.:
Properties of pmfs:

Nonnegativity:

Total probability:

Relation to cdf:
Continuous r.v.s
A r.v.
is said to be continuous is there is a function
such that
for all . In this case, is an absolutely continuous function. ^{ 1 }
Properties:
 .
called the probability density function (pdf) of .
 .
 for all .

.
Properties of Gaussian Random Variables
Let .
 If then
 .
 .
 A Gaussian r.v. is completely characterized by its first two moments (i.e., the mean and variance).
Bibliography
 P. Billingsley, Probability and Measure . New York: Wiley, 1986.