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Probability and Statistics
Dr. Saeid Moloudzadeh
Axioms of Probability/Basic Theorems
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• 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 be a mutually exclusive set of events. Then
1 2, , , nA A A
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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,
where N(A) is the number of points of A.
( )
N AP A
N
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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.
