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Probability and Statistics Axioms of Probability/ Basic Theorems • • • • • Dr. Saeid Moloudzadeh www.soran.edu.iq • • • • 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 1 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 www.soran.edu.iq 2 Chapter 1: Axioms of Probability Context • Sample Space and Events • Axioms of Probability • Basic Theorems www.soran.edu.iq 3 Chapter 1: Axioms of Probability Context • Sample Space and Events • Axioms of Probability • Basic Theorems www.soran.edu.iq 4 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. www.soran.edu.iq 5 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. www.soran.edu.iq 6 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 www.soran.edu.iq 7 Section 3: Axioms of Probability www.soran.edu.iq 8 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. www.soran.edu.iq 9 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. www.soran.edu.iq 10 Section 3: Axioms of Probability www.soran.edu.iq 11 Section 3: Axioms of Probability www.soran.edu.iq 12 Section 4: Basic Theorems www.soran.edu.iq 13 Section 4: Basic Theorems www.soran.edu.iq 14 Section 4: Basic Theorems www.soran.edu.iq 15 Section 4: Basic Theorems www.soran.edu.iq 16 Section 4: Basic Theorems www.soran.edu.iq 17