Stochastic modeling of any type of system needs basic knowledge of probability theory. It is my experience that a brief summary about the most important concepts and theorems is very useful because the readers might have different level approaches to the theory of probability. The refresher concentrate only on those theorems and distributions which are closely related to this material. It should be noted that there are many good textbooks about theory of probability in all over the word. Moreover, a number of digital versions can be downloaded from the internet, too. I would recommend any of the following books, Allen [ 2 ], Gnedenko et.al. [ 29 ], Jain [ 41 ], Kleinrock [ 48 ], Kobayashi and Mark [ 51 ], Ovcharov and Wentzel [ 60 ], Ravichandran [ 64 ], Rényi [ 66 ], Ross [ 67 ], Stewart [ 74 ], Tijms [ 91 ], Trivedi [ 94 ].
Theorem 1.1.
(Basic Forms of the Law of Total Probability)
Let 
be a set of mutually exclusive exhaustive events
with positive probabilities and let 
be any event. Then
where
Theorem 1.2.
(Bayes' Theorem or Bayes' Rule)
Let
be a set of mutually exclusive exhaustive events
with positive probabilities and let 
be any event of positive probability.
Then
Definition 1.3.
Let
, 
, the distribution of a discrete random variable
. The mean ( first moment, expectation, average )
of 
is defined as 
if this series is absolute convergent. That is the
mean of 
is
Definition 1.4.
Let
be the density function of a continuous random
variable 
. If 
is finite then the mean is defined
by
Without proof the main properties of the expectation are as follows
If 
, then
exists and 
,
exists and 
,
exists and 
, provided 
are 
are independent
exists and 
,
exists and 
, if the second moments are exist,If
, then 
, 
.
Theorem 1.5. (Theorem of Total Moments) The most commonly used forms are
where 
denotes the 
th conditional moment. The continuous version
is
In case of 
we have the theorem of total
expectation.
Definition 1.6.
(Variance)
Let 
be a random variable with a finite mean
. Then
is called the variance of 
provided it is finite.
The following properties hold
Ha
then 
.
for any 
.
; 
if and only if 
.
Definition 1.7.
(Squared coefficient of variation)
The coefficient 
is defined as the squared
coefficient of variation of random variable 
.