Download Axioms of Probability

Probability and Statistics
Axioms of Probability/
Basic Theorems
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Dr. Saeid Moloudzadeh
www.soran.edu.iq
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Contents
Descriptive Statistics
Axioms of Probability
Combinatorial Methods
Conditional Probability and
Independence
Distribution Functions and
Discrete Random Variables
Special Discrete Distributions
Continuous Random Variables
Special Continuous Distributions
Bivariate Distributions
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Probability and Statistics
Contents
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Descriptive Statistics
Axioms of Probability
Combinatorial Methods
Conditional Probability and Independence
Distribution Functions and Discrete Random Variables
Special Discrete Distributions
Continuous Random Variables
Special Continuous Distributions
Bivariate Distributions
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Chapter 1: Axioms of Probability
Context
• Sample Space and Events
• Axioms of Probability
• Basic Theorems
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Chapter 1: Axioms of Probability
Context
• Sample Space and Events
• Axioms of Probability
• Basic Theorems
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Section 3: Axioms of Probability
Definition 2-2-1 (Probability Axioms): Let S be
the sample space of a random phenomenon.
Suppose that to each event A of S, a number
denoted by P(A) is associated with A. If P
satisfies the following axioms, then it is called
a probability and the number P(A) is said to be
the probability of A.
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Section 3: Axioms of Probability
Let S be the sample space of an experiment. Let A and B be
events of S. We say that A and B are equally likely if P(A) = P(B).
We will now prove some immediate implications of the axioms
of probability.
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Section 3: Axioms of Probability
Theorem 1.1: The probability of the empty set
 is 0. That is, P( ) = 0.
Theorem 2-2-3: Let A1 , A2 , , An  be a
mutually exclusive set of events. Then
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Section 3: Axioms of Probability
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Section 3: Axioms of Probability
It is now called the classical definition of probability. The
following theorem, which shows that the classical definition is
a simple result of the axiomatic approach, is also an important
tool for the computation of probabilities of events for
experiments with finite sample spaces.
Theorem 1.3: Let S be the sample space of an experiment. If S
has N points that are all equally likely to occur, then for any
event A of S,
N A 
P (A ) 
N
where N(A) is the number of points of A.
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Section 3: Axioms of Probability
Example 1.11: Let S be the sample space of flipping
a fair coin three times and A be the event of at
least two heads; then
S ={HHH,HTH,HHT, HTT,THH, THT, TTH, TTT}
and A = {HHH,HTH,HHT,THH}. So N = 8 and N(A) = 4.
Therefore, the probability of at least two heads in
flipping a fair coin three times is N(A)/N = 4/8 =
1/2.
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Section 3: Axioms of Probability
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Section 3: Axioms of Probability
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Section 4: Basic Theorems
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Section 4: Basic Theorems
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Section 4: Basic Theorems
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Section 4: Basic Theorems
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Section 4: Basic Theorems
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