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
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller carinewildeboer. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $35.02. You're not tied to anything after your purchase.
