CHAPTER 2 A REVIEW OF BASIC STATISTICAL CONCEPTS
CHAPTER 2 A REVIEW OF BASIC STATISTICAL CONCEPTS
2.1 SOME NOTATION 1. The summation Notation 多x, 答、 can be abbreviated as:∑x,or∑x 2. Properties of the Summation Operator kX=k>x ∑(X1+Y)=∑X+∑Y ∑(a+bX1)=m+b∑X
2.1 SOME NOTATION • 1. The Summation Notation can be abbreviated as: or • 2. Properties of the Summation Operator = = = + + + i n i Xi X X X n 1 1 2 = − i n i Xi 1 Xi X X = = n i k nk 1 kXi = kXi Xi +Yi =Xi +Yi ( ) a + bXi = na + bXi ( )
2.2 EXPERIMENT SAMPLE SPACE SAMPLE POINT AND EVENTS 1.E× periment A statistical/ random experiment: a process leading to at least two possible outcomes with uncertainty as to which will occur 2. Sample space or population The population or sample space: the set of all possible outcomes of an experiment 3. Sample Point Sample point each member, or outcome, of the sample space(or population)
2.2 EXPERIMENT, SAMPLE SPACE, SAMPLE POINT, AND EVENTS 1. Experiment A statistical/random experiment: a process leading to at least two possible outcomes with uncertainty as to which will occur. 2.Sample space or population The population or sample space: the set of all possible outcomes of an experiment 3. Sample Point Sample Point : each member, or outcome, of the sample space (or population)
2.2 EXPERIMENT SAMPLE SPACE SAMPLE POINT AND EVENTS 4。E∨ents An event: a collection of the possible outcomes of an experiment; that is, it is a subset of the sample space Mutually exclusive events the occurrence of one event prevents the occurrence of another event at the same time Equally likely events: one event is as likely to occur as the other event Collectively exhaustive events: events that exhaust all possible outcomes of an experiment
2.2 EXPERIMENT, SAMPLE SPACE, SAMPLE POINT, AND EVENTS • 4. Events An event: a collection of the possible outcomes of an experiment; that is, it is a subset of the sample space. Mutually exclusive events: the occurrence of one event prevents the occurrence of another event at the same time. Equally likely events: one event is as likely to occur as the other event. Collectively exhaustive events: events that exhaust all possible outcomes of an experiment
2. 3 RANDOMVARIABLES o A random/stochastic variable(rv. for short):a variable whose (numerical) value is determined by the outcome of an experiment. (1)A discrete random variable --an rv. that takes on only a finite (or accountably infinite) number of values (2)A continuous random variable -=an rv that can take on any value in some interval of values
2.3 RANDOM VARIABLES • A random/stochastic variable（r.v., for short）: a variable whose (numerical) value is determined by the outcome of an experiment. • （1）A discrete random variable—— an r.v. that takes on only a finite (or accountably infinite) number of values. • （2）A continuous random variable——an r.v. that can take on any value in some interval of values
2. 4 PROBABILITY The classical or a priori definition if an experiment can result in n mutually exclusive and equally likely outcomes and if m of these outcomes are favorable to event a then P(A, the probability that A occurs, is m/n P(4)= the number of outcomes favorable to a the total number of outcomes Two features of the probability (1) The outcomes must be mutually exclusive: (2) Each outcome must have an equal chance of occurring
2.4 PROBABILITY • 1．The Classical or A Priori Definition：if an experiment can result in n mutually exclusive and equally likely outcomes, and if m of these outcomes are favorable to event A, then P(A), the probability that A occurs, is m/n Two features of the probability: （1）The outcomes must be mutually exclusive; （2）Each outcome must have an equal chance of occurring. the total number of outcomes the number of outcomes favorable to A ( ) = = n m P A
2. 4 PROBABILITY 2. Relative Frequency or Empirical Definition Frequency distribution how an r.v. are distributed Absolute frequencies: the number of occurrence of a gIven event. Relative frequencies: the absolute frequencies divided by the total number of occurrence Empirical Definition of Probability if in n trials (or observations), m of them are favorable to event then P(A), the probability of event A, is simply the ration m/n (that is, relative frequency) provided n, the number of trials, is sufficiently large In this definition we do not need to insist that the outcome be mutually exclusive and equally likely
2.4 PROBABILITY • 2.Relative Frequency or Empirical Definition Frequency distribution: how an r.v. are distributed. Absolute frequencies: the number of occurrence of a given event. Relative frequencies: the absolute frequencies divided by the total number of occurrence. Empirical Definition of Probability: if in n trials（or observations）, m of them are favorable to event A, then P（A）, the probability of event A, is simply the ration m/n, （that is, relative frequency）provided n, the number of trials, is sufficiently large In this definition, we do not need to insist that the outcome be mutually exclusive and equally likely
2.4 PROBABILITY 3. Properties of probabilities (1)0≤P(A)≤1 (2) If A, B, Cr.. are mutually exclusive events then P(A+B+C+.=P(A+P(B)+P(C)+ (3 If A, B, Cr.. are mutually exclusive and collectively exhaustive set of events, P(A+B+C+…)=R(A+P(B)+P(C)+.=1
2.4 PROBABILITY • 3. Properties of probabilities (1) 0≤P(A)≤1 (2) If A, B, C, ... are mutually exclusive events, then: P(A+B+C+...)=P(A)+P(B)+P(C)+... (3) If A, B, C, ... are mutually exclusive and collectively exhaustive set of events, P(A+B+C+...)=P(A)+P(B)+P(C)+...=1
2.4 PROBABILITY Rules of probability 1)If A, B, Cr, are any events they are said to be statistically independent events if: P(ABCD=P(AP(BP(C 2)If events a B C.. are not mutually exclusive, then P(A+B)=P(A)+P(B)P(AB) Conditional probability of A, given B P(|) PCB) (B) Conditional probability of B, given A (B| PCB) 1
2.4 PROBABILITY Rules of probability: • 1) If A, B, C,...are any events, they are said to be statistically independent events if: P(ABC...)=P(A)P(B)P(C) • 2) If events A, B, C, ... are not mutually exclusive, then P(A+B)=P(A)+P(B)－P(AB) Conditional probability of A, given B Conditional probability of B, given A ( ) ( ) ( | ) P B P AB P A B = ( ) ( ) ( | ) P A P AB P B A =
2.5 RANDOMVARIABLES AND PROBABILITY DISTRIBUTION FUNCTION (PDF) he probability distribution function or probability density function (PDF)of a random variable X: the values taken by that random variable and their associated probabilities 1. PDF of a discrete random variable Discrete rv. (x takes only a finite (or countably infinite) number of values various values of the random variable /o d Probability distribution or probability density function (PDf)-it shows how the probabilities are spread over or distributed over the PDF of a discrete r.v.X f(X)=P(X=X) for i=1, 2, 3., n And 0 forX≠X
2.5 RANDOM VARIABLES AND PROBABILITY DISTRIBUTION FUNCTION (PDF) • The probability distribution function or probability density function (PDF) of a random variable X: the values taken by that random variable and their associated probabilities. • 1.PDF of a Discrete Random Variable Discrete r.v.（X） takes only a finite（or countably infinite） number of values. • Probability distribution or probability density function (PDF)—it shows how the probabilities are spread over or distributed over the various values of the random variable X. • PDF of a discrete r.v.(X) f(X)= P(X=Xi ) for i=1,2,3...,n And 0 for X≠Xi