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
Voordelen van het kopen van samenvattingen bij Stuvia op een rij:
√ Verzekerd van kwaliteit door reviews
Stuvia-klanten hebben meer dan 700.000 samenvattingen beoordeeld. Zo weet je zeker dat je de beste documenten koopt!
Snel en makkelijk kopen
Je betaalt supersnel en eenmalig met iDeal, Bancontact of creditcard voor de samenvatting. Zonder lidmaatschap.
Focus op de essentie
Samenvattingen worden geschreven voor en door anderen. Daarom zijn de samenvattingen altijd betrouwbaar en actueel. Zo kom je snel tot de kern!
Veelgestelde vragen
Wat krijg ik als ik dit document koop?
Je krijgt een PDF, die direct beschikbaar is na je aankoop. Het gekochte document is altijd, overal en oneindig toegankelijk via je profiel.
Tevredenheidsgarantie: hoe werkt dat?
Onze tevredenheidsgarantie zorgt ervoor dat je altijd een studiedocument vindt dat goed bij je past. Je vult een formulier in en onze klantenservice regelt de rest.
Van wie koop ik deze samenvatting?
Stuvia is een marktplaats, je koop dit document dus niet van ons, maar van verkoper carinewildeboer. Stuvia faciliteert de betaling aan de verkoper.
Zit ik meteen vast aan een abonnement?
Nee, je koopt alleen deze samenvatting voor €30,49. Je zit daarna nergens aan vast.
