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In The Name of God. Malayer University. Department of Civil Engineering. PROBABILITY AND STATISTICS FOR ENGINEERS. Taught by:. Dr. Ali Reza Bagherieh. Introduction. - PowerPoint PPT Presentation
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PROBABILITY AND STATISTICS FOR ENGINEERS
In The Name of God
Taught by:
Dr. Ali Reza Bagherieh
Malayer University
Department of Civil Engineering
Introduction
This course is concerned with the development of basic principles
in constructing probability models and the subsequent analysis of
these models.
Probability and Random Variables
Basic Probability Concepts
In studying a random phenomenon, we are dealing with an experiment
of which the outcome is not predictable in advance.
Example: Games of chance
Physical or natural phenomena involving uncertainties
Uncertainty comes from : 1-complexity 2-lack of understanding of
all the causes and effects 3-lack of informationFor example,
weather prediction
The second class of problems widely studied by means of
probabilisticmodels concerns those exhibiting variability.
for example, a problem in traffic flow where an engineer wishes to
know the number of vehicles crossing a certain point on a road
within a specified interval of time.
ELEMENTS OF SET THEORY
A set is a collection of objects possessing some common
properties
These objects are called elements of the set and they can be of any
kind with anyspecified properties.
A set containing no elements is called an empty or null set and is
denoted by
finite sets ------infinite sets
enumerable or countable set
All of its elements can be arranged in such a way that there is a
one-to-one correspondence between them and all positive
integers
nonenumerable or uncountable
If every element of a set A is also an element of a set B, the set
A is calleda subset of B and this is represented symbolically
by
The largest set containing all elements of all the sets under
consideration is called space and is denoted by the symbol
S.
If AB =, sets A and B contain no common elements, and we call A
and B disjoint.
S AM P L E S P A CE AN D P RO BA BILIT Y M E AS U RE
In probability theory, we are concerned with an experiment with an
outcomedepending on chance, which is called a random
experiment.
It is assumed that all possible distinct outcomes of a random experiment are known and that they are elements of a fundamental set known as the sample space.
Each possible outcome is called a sample point,
An event is generally referred to as a subset of the sample
space having one or more sample points as its elements.
For a given random experiment, the associated sample space is not
unique Its construction depends upon the point of view adopted as
well as the questions to be answered.
All relations between outcomes or events in probability theory
can be described by sets and set operations.
Disjoint sets are mutually exclusive events in probability
theory.
Given a random experiment, a finite number P(A) is assigned to
every event A in the sample space S of all possible events. The
number P(A) is a function of set A and is assumed to be defined for
all sets in S. It is thus a set function, and P(A) is called
theprobability measure of A or simply the probability of
A.
ASSIGNMENT OF PROBABILITY
For problems in applied sciences, a natural way to assign the
probability of an event is through the observation of relative
frequency.
Assuming that a random experiment is performed a large number of
times, say n,then for any event A let nA be the number of
occurrences of A in the n trials anddefine the ratio nA /n as the
relative frequency of A.
Under stable or statistical regularity conditions, it is expected
that this ratio will tend to a unique limit as n becomes
large.
Another common but more subjective approach to probability
assignment isthat of relative likelihood. When it is not feasible
or is impossible to perform anexperiment a large number of times,
the probability of an event may be assignedas a result of
subjective judgment.
The statement there is a 40% probability of rain tomorrow is an
example in this interpretation, where the number 0.4 is assigned on
the basis of available information and professional
judgment.
STATISTICAL INDEPENDENCE
Given individual probabilities P(A) and P(B) of two events A and B,
what is P(AB)?
A special case in which the occurrence or nonoccurrence of one does
not affect the occurrence or nonoccurrence of the other. In this
situation events A and B are called statistically independent or
simply independent
CONDITIONAL PROBABILITY
Given two events A and B associated with a random experiment,
P(A/B) probability is defined as the conditional probability of A,
given that B has occurred.
Conditional probabilities are probabilities
They satisfy the probability axioms.
As for the third axiom, if A1, A2, . . . are mutually exclusive,
then A1B, A2B, . . .are also mutually exclusive. Hence,
Theorem 2. 1: theorem of total probability . Suppose that events
B1, B2, . . . , andBn are mutually exclusive and exhaustive (i.e. S
B1 B2 Bn). Then,for an arbitrary event A,
Random Variables and Probability Distributions
We can make statements concerning the events that can occur, and
these statementsare made based on probabilities assigned to simple
outcomes.
A systematic and unified procedure is needed to facilitate making
these statements
Each of the possible outcomes of a random experiment be represented
by a real number.
Each outcome is identified by its assigned real number rather than
by its physical description.
A sample space of arbitrary elements by a new sample space having
only real numbers as its elements
Use arithmetic means for probability calculations.
RANDOM VARIABLES
We assume that it is possible to assign a real number X(s) for each
outcome s following a certain set of rules.
The number X(s) is really a real-valued point function defined over
the domain of the basic probability space
Let us again assign number one to the event success and zero to
failure. If X is the random variable associated with this
experiment, then takes on two possible values: 1 and 0.
The random variable X is called a random variable if it is defined
over a sample space having a finite or a countably infinite number
of sample points. In this case, random variable takes on discrete
values, and it is possible to enumerate all the values it may
assume.
In the case of a sample space having an uncountably infinite number
of sample points, the associated random variable is called a
continuous random variable, with its values distributed over one or
more continuous intervals on the real line.
We make this distinction because they require different probability
assignment considerations.
We note here that an analysis involving random variables is
equivalent to considering a random vector having the random
variables as its components.
PROBABILITY DISTRIBUTIONS
The behavior of a random variable is characterized by its
probability distribution,that is, by the way probabilities are
distributed over the values it assumes.
A probability distribution function and a probability mass
function are twoways to characterize this distribution for a
discrete random variable. They areequivalent in the sense that the
knowledge of either one completely specifiesthe random
variable.
Probability Distribution Function
Important properties possessed by a PDF
completely characterizes random variable
PROBABILITY DENSITY FUNCTION FOR CONTINUOUS RANDOM
VARIABLES
Dr. Ali Reza Bagherieh
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Axiom
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