A summary containing all theorems, propositions and other important subjects from the book 'Introduction to Probability Models' chapters 1-7 and chapter 9. This is the exactly what students from the RUG need to study for the exam of Stochastic Models.
Chapter 1 · Introduction to Probability Theory
1.2 Sample Space and Events
Sample space, S = {...}: set of all possible outcomes of an experiment.
Event, E = {...}: any subset of the sample space S.
Union, E ∪ F : either in E or F or in both E and F.
Intersection, E ∩ F : all outcomes that are both in E and in F.
Mutually exclusive, EF = ∅: the event consisting of no outcomes. E and F have zero overlap.
S∞
Union of multiple events, n=1 En : the event that consists of all outcomes that are in En for at least
one value of n = 1, 2, .... T∞
Intersection of multiple events, n=1 En : the event consisting of those outcomes that are in all of the events
En , n = 1, 2, ...
Complement, E c : consists of all outcomes in the sample space that are not in E.
1.3 Probabilities Defined on Events
Probability of event E, P (E): for each event E of sample space S, we assume P(E) exists and satisfies the
following:
(i) 0 ≤ P (E) ≤ 1
(ii) P (S) = 1
(iii) For any sequence of events E1 , E2 , ... that are mutually exclusive, that is, En Em = ∅ when n ̸= m, then:
∞
[ ∞
X
P( En ) = P (En )
n=1 n=1
Inclusion-exclusion identity: the probability of the union of n events equals the sum of the probabilities of
these events taken one at a time minus the sum of the probabilities of these events taken two at a time plus the
sum of the probabilities of these events taken three at a time, and so on. For any n events E1 , E2 , E3 , ..., En
X X X
P (E1 ∪ E2 ∪ · · · ∪ En ) = P (Ei ) − P (Ei Ej ) + P (Ei Ej Ek )
i i<j i<j<k
X
− P (Ei Ej Ek El )
i<j<k<l
+ · · · + (−1)n+1 P (E1 E2 · · · En )
1.4 Conditional Probabilities
Conditional Probability, P (E|F ): the probability that E occurs, given that F occurs:
P (EF )
P (E|F ) =
P (F )
1
,1.5 Independent Events
Independence: when the occurrance of F has no effect on E, E and F are independent if:
P (EF ) = P (E)P (F ) ⇒ P (E|F ) = P (E)
Or for multiple events:
P (E1′ , E2′ , ..., Er′ ) = P (E1′ )P (E2′ ) · · · P (Er′ )
Independent trials: sequence of experiments, each of which results in either ”success” or ”failure”, that are
independent:
n
Y
P (Ei1 Ei2 · · · Ein ) = P (Eij )
j=1
1.6 Bayes’ Formula
The probability of the event E is a weighted average of the conditional probability of E given that F has occurred
and the conditional probability of E given that F has not occurred:
P (E) = P (E|F )P (F ) + P (E|F c )(1 − P (F ))
Bayes’ Formula:
P (E|Fj )P (Fj )
P (Fj |E) = Pn
i=1 P (E|Fi )P (Fi )
Chapter 2 · Random Variables
2.1 Random Variables
Random variables: real-valued functions defined on the sample space, can be discrete or continuous.
Indicator random variable for event E:
(
1, if E occurs
I=
0, if E does not occur
Cumulative Distribution Function (cdf ) F (b): the probability that random variable X takes on value less
or equal to b:
F (b) = P (X ≤ b)
Its properties are:
(i) F (b) is a non-decreasing function of b
⇒ P (a < X ≤ b) = F (b) − F (a) ∀a < b,
(ii) limb→∞ F (b) = F (∞) = 1
(iii) limb→−∞ F (b) = F (−∞) = 0
Probability X is strictly smaller than b:
P (X < b) = lim P (X ≤ b − h) = lim F (b − h)
h→0+ h→0+
2.2 Discrete Random Variables
Discrete: random variable can take on at most a countable number of possible values.
Probability mass function, p(a) of X:
p(a) = P (X = a)
2
, If X must assume one of the values x1 , x2 , ..., then:
p(xi ) > 0, i = 1, 2, ...
p(x) = 0, all other values of x
∞
X
Therefore, we have: p(xi ) = 1
i=1
The cdf F can be expressed as: X
F (a) = p(xi )
allxi ≤a
2.2.1 The Bernoulli Random Variable
Trial with either ”success” (X = 1) or ”failure” (X = 0). The pmf function of a Bernoulli random variable X
is given by:
p(0) = P (X = 0) = 1 − p
p(1) = P (X = 1) = p
for some p ∈ (0, 1).
2.2.2 The Binomial Random Variable
If X represents the number of successes in n trials, it has a binomial pmf having parameters (n, p):
n i
p(i) = p (1 − p)n−i , i = 0, 1, ..., n
i
where:
n n!
=
i (n − 1)!i!
2.2.3 The Geometric Random Variable
Independent trials are performed until a success (with probability p occurs. Geometric random variable X is
the number of trials until the first success, the pmf is given by:
p(n) = P (X = n) = (1 − p)n−1 p, n = 1, 2, ...
2.2.4 The Poisson Random Variable
Random variable X is Poisson distributed with parameter λ > 0 if its pmf is:
λi
p(i) = P (X = i) = e−λ , i = 0, 1, ...
i!
Can be used to approximate binomial random variable if n is large and p is small (use λ = np).
2.3 Continuous Random Variables
Probability Density Function, pdf, f (x): a non-negative function, defined for all real x ∈ (−∞, ∞), having
the property that for any set B of real numbers:
Z
P (X ∈ B) = f (x)dx
B
We obtain: Z b
P (a ≤ X ≤ b) = f (x)dx
a
Z a
d
F (a) = P (X ∈ (−∞, a]) = f (x)dx ⇒ F (a) = f (a)
−∞ da
3
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