Probability and Statistics Dr. Saeid Moloudzadeh Axioms of Probability/ Basic Theorems 1 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

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 2

Chapter 1: Axioms of Probability Context Sample Space and Events Axioms of Probability Basic Theorems 3

Chapter 1: Axioms of Probability Context Sample Space and Events Axioms of Probability Basic Theorems 4

Section 3: Axioms of Probability Definition (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. 5

Section 3: Axioms of Probability 6 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.

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 7

Section 3: Axioms of Probability 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, where N(A) is the number of points of A. 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. 10

Section 3: Axioms of Probability 11

Section 3: Axioms of Probability 12

Section 4: Basic Theorems 13

Section 4: Basic Theorems 14

Section 4: Basic Theorems 15

Section 4: Basic Theorems 16

Section 4: Basic Theorems 17