Theory of probability and mathematical statistics - …

1 Theory of probability and mathematical statisticsTom a s Mrkvi ckaBibliography[1] J. And d z l:Matematick d z statistika, SNTL/ALFA, Praha 1978[2] J. And d z l:Statistick d z metody, Matfyzpress, Praha 1998[3] V. Dupa d z , M. Hu d z kov d z :Pravd d z podobnost a matematick d z statis-tika, Karolinum, Praha 1999[4] d z ka, d z d z kov d z : d z voddoteoriepravd d z podobnosti,PF JU, d z esk d z Bud d z jovice 2008[5] T. Mrkvi d z ka, V. Petr d z d z kov d z : d z vod do statistiky,PF JU, d z esk d z Bud d z jovice 20061 Chapter 1 probability theory2 CHAPTER 1.

2 probability Definition of probabilityDefinition Classical definition:LetA1, .. ,Anbe random events,such that every time one and only one random event happen, all the event are equally let the eventAhappen if happen one of the eventAi1, .. ,Aik. Then theprobability ofAisP(A) = Kolmogorov definition of probabilityProbability space ( ,A,P). is nonempty set of all results of the random experiment, results are .Ais -algebra on . (The set of all nice subsets of )P:A Ris function giving every setA Aits probability (0 P(A) 1,P( ) = 1).

3 This is probability 1. probability THEORY4 RulesP(AC) = 1 P(A)P(A B) =P(A) +P(B),if A, B are disjointP(A B) =P(A) +P(B) P(A B)CHAPTER 1. probability THEORY5 Examples1. A phone company found that 75% of customers want text messaging, 80%photo capabilities and 65% both. What is the probability that customer willwant at least one of these?2. What is the probability , that there exists two students, in a class withnstu-dents, who have the birth dates in a same Two boats, which uses same harbor, can enter whenever during 24 hours.

4 Theyenter independently. The first boat stays 1 hour, second 2 hours. What is theprobability, that no boat will has to wait to enter the Student choose 3 questions from 30 during an exam. There are 10 questionsfrom algebra, 15 from analysis and 5 from 5 geometry. What is the probability ,that he choose at least two questions from the same 1. probability Conditional probabilityDefinition ( ,A,P) is probability space andA,Bare random events,whereP(B)>0. We define the conditional probability ofAunder the condi-tionBby relationP(A|B) =P(A B)P(B).

5 ( ) IndependenceConsider now two random eventsAandB. If the following holdsP(A|B) =P(A)a P(B|A) =P(B),( )then we speak about its independence. From ( ) wee see, that the probability ofAunder the conditionBdo not depend onBand vica versa. From ( ) and definitionof conditional probability we have the following definition of the 1. probability THEORY7 Definition eventsAandBare independent, ifP(A B) =P(A) P(B).( )Theorem ,..,Anare independent. ThenP( ni=1Ai) = 1 n i=1[1 P(Ai)].( )Rule:WhenAandBare not independent, one can use:P(A B) =P(A|B)P(B).

6 CHAPTER 1. probability THEORY8 Examples1. A phone company found that 75% of customers want text messaging, 80%photo capabilities and 65% both. What are the probabilities that a personwho wants text messaging also wants photo capabilities and that a person whowants photo capabilities also wants text messaging?2. The players A and B throw a coin and they alternate. A starts, then B, thenA, .. The game ends when the first one obtain head on the coin. What is theprobability of winning of A and The probability that one seed grow up is You have 10 seeds.

7 What isthe probability that exactly 1 seed grow up. (2 seeds, ..) What is the mostprobable result?4. The probability that one seed grow up is How many seeds you have to use,to have 99% probability , that at least 1 seed will grow 1. probability Random variablesDefinition Random variableis every measurable mappingXfrom( ,A,P) Distribution functionFof a random variableXis givenbyF(x) =P( :X( ) x).ShortlyF(x) =P(X x).CHAPTER 1. probability THEORY10 Discrete random variablesRandom variableXcan have maximally countably many valuesx1,x2.

8 P(X=xi) =pi,i= 1,2,.. pi= :P(X B) = i:xi ixipi.( )Expectation of a function ofXEg(X) = ig(xi)pi.( )CHAPTER 1. probability THEORY11 Basic distributions of discrete random variablesAlternativ distribution A(p)represents success/unsuccess of the experiment0< p < (X= 1) =p, P(X= 0) = 1 ,Var(X) =p(1 p).Binomial distribution Bi(n,p)represents number of successes innindepen-dent experiments. The probability of success is 0< p <1. In other words, binomialdistribution is sum ofnindependent alternative (X=k) =(nk)pk(1 p)n k, k= 0,1.

9 , ,Var(X) =np(1 p).Hypergeometric distribution HGeom(n,M,N)is used instead of Binomialin experiments, wherenrepresents number of draws without returning (Binomial-nis with returning.) from box which hasNelements, andMelements representCHAPTER 1. probability THEORY12success. (BinomialM/N=p.) Hypergeometrical distribution then representsnumber of success in this (X=k) =(Mk)(N Mn k)(Nn), k= 0,1,.., ,Var(X) =nMN(1 MN)N nN distribution Po( ) >0 represents number of events which appearin time of (X=k) =e kk!

10 EX= ,Var(X) = .Geometrical distribution Geom(p)represents number of experiment untilfirst success appears. The probability of success is 0< p < (X=k) =p(1 p) 1. probability THEORY13EX=1 pp,Var(X) =1 1. probability THEORY14 Examples1. The phone central connects 15 talks during one hour in average. What is theprobability that during 4 minutes it connects: a) exactly one talk, b) at leastone talk, c) at least two talks and in maximum 5 The man phone to phone central during the maximal load, when there is prob-ability to be connected.