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# Introduction to Mathematical Statistics I ST 421

NCS

GPA 3.79

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This 73 page Class Notes was uploaded by Jordane Kemmer on Thursday October 15, 2015. The Class Notes belongs to ST 421 at North Carolina State University taught by Staff in Fall. Since its upload, it has received 17 views. For similar materials see /class/223957/st-421-north-carolina-state-university in Statistics at North Carolina State University.

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Date Created: 10/15/15

Probability of an Eventl Steps to nd the probability of an event 1 De ne the experiment and clearly determine how to describe one simple event List the simple events associated With the experiment and test each to make certain that it cannot be decomposed This de nes the sample space S Assign reasonable probabilities to the sample points in S making certain that De ne the event of interest A as a speci c collection of sample points A sample point is in A if A occurs When the sample point occurs Test all sample points in S to identify those in A Find PA by summing the probabilities of the sample points in A These steps will be illustrated in Class with examples Counting sample points I If a sample space contains N equiprobable sample points and an event A contains exactly na sample points then PA 2 71a N We present now some useful results from the theory of combinatorial analysis mm rule With m elements 01412 am and n elements 6162 6 it is possible to form mm 2 m X n pairs containing one element from each group proof We could form a table like in Figure 29 to indicate the number of pairs There would one square in the table for each ai bj pair and hence a total of m X n squares The following theorem can be used to determine the number of ordered arrangements that can be formed An ordered arrangement of 7 distint objects is called a permutation The number of ways of ordering n distinct objects taken 7 at a time Will be designated by the symbol P731 The next result is used to determine the number of subsets of various sizes that can be formed by partitioning a set of n distinct objects into k nonoverlapping groups Theorem The number of ways of partioning n distinct objects into k distinct groups containing n1 n2 nk objects respectively Where each object appears in exactly one group and 21 n n is n n N 2 v v v n1n2nk n1n2 nk proof presented in class see book page 43 The terms 71 771712 nk are called multinomial coef cients because they occur in the expansion of the multinomial term 31 32 yk raised to the nth power 311y2ykquot n 1 2 W 31 32 nk Wheren1n2nkn De nition The number of combinations of 77 objects taken 7 at a time is the number of subsets each of size 7 that can be formed from the 77 objects This number Will be denoted by C or Theorem The number of unordered subsets of size 7 chose Without replacement from 77 available objects is 71 nP1 77 7a 7 7 7 7z 7 proof page 45 n The terms are generally referred to as r binomial coef cients because they occur in the binomial expansion TL yquot i0 7 it n zz my The event composition methodl The event composition method for calculating the probability of an event A expresses Z as a composition involving unions and or intersections of other events The laws of probability are then applied to nd PA Steps in the event composition method 1 De ne the experiment 2 visualize the nature of the sample points Identify a few to clarify your thinking 3 Write an equation expressing the event of interest say A as ac omposition of two or more events using unions intersections and or complements Make certain that event A and the event implied by the composition represent the same set of sample points 4 Apply the additive and multiplicative laws of probability to the compositions obtained in step 3 to nd PA Bayes Rule I The event composition approach is sometimes facilitated by Viewing the sample space S as a union of mutually exclusive subsets a partition De nition For some positive integer k let the sets 8182 Bk be such that 1 SzBluBQUUBk 2 then the collection of sets 131 82 Bk is said to be a partition of S Theorem Assume that 81 Bg Bk is a partition of S such that PBi gt 0 for 239 12k Then for any event A k PA ZPAIBzPBz i1 Bayes Rule Assume that 81132 Bk is a partition of S such that PBi gt 0 for i 2 12k Then PABjPBj HEBIA W Fall 2001 130 220 MWF Harrelson 320 INTRO TO MATHEMATICAL STATISTICS I Montserrat Fuentes Statistics Department N CS U fuentes statncsuedu http wwwstatncsuedu Nfuentes Introduction I What is Statistics 0 Introduction 0 Graphical methods 0 Numerical methods 0 Inference 0 Theory and Reality Population The large body of data that is the target of our interest eg collection of all GPA s of US students in 2000 Sample subset selected from a population eg a collection of GPA s representing only two scores from each state of US We are forced to look at samples primarily for the following reasons Economy Timeliness Large populations Inaeeessibility Statistics The objective of statistics is to make an inference about a population based on information contained in a sample from that population and to provide an associated measure of goodness for the inference Examples 0 sample of voters to estimate the true fraction of all voters Who favor a particular candidate 0 A decision is made regarding the relative merits of two manufacturing processes based on examination of samples of products Graphical Methods sort the data into groups de ned by possible variable values Then count how many data items were sorted into each category Frequency The count for a particular category Relative frequency The decimal value obtained by dividing the frequency by total number of data items The graph is constructed by subdividing the axis of measurements into intervals of equal Width histogram NUMERICAL METHODS Measures of central tendency MEAN The mean of a sample of n measured responses 31 32 yn is given by 1 n g a 312 21 The corresponding population mean is denoted u MEDIAN The value above and below an equal number of observation MODE Most frequently occurring value Measures of dispersion VARIANCE The variance of a sample is the sum of the square of the differences between the measurements and their mean divided by n 1 TL 1 2 2 2 i1 The corresponding population variance is denoted by the symbol 02 the larger the variance the greater the amount of variation Within the set of observations THE STANDARD DEVIATION It is the positive square root of the variance RANGE Difference between the largest and the smallest values For data mound shaped EMPIRICAL RULE For a distribution of measurements that is approximately normal bell shaped it follows that the interval With endpoints V i 0 contains approximately 68 of the measurements V i 20 contains approximately 95 of the measurements V i 30 contains approximately 997 of th measurements INFERENTIAL STATISTICS Studies of the properties of a small sample of data selected from a much larger body of data population and attempts using mathematics probability to infer the nature of the same properties for the population Probability is the mechanism used in making statistical inferences Intuitive assessments of probabilities are unsatisfactory and we need a rigorous theory of probability in order to develop methods of inference THEORY AND REALITY We Will work with theoretical or mathematical models for acquiring and utilizing information in real life The process of nding a good model is not simple and usually requires several simplifying assymptions unifrom string mass no air resistance etc The model Will not be an exact representation of nature Momentgenerating functions I This method is based on a uniqueness theorem which states that if two rV have identical moment generating functions the two rV s possess the same probability distributions To use this method we must nd the moment generating function for U and compare it with the moment generating functions for the common discrete and continuous rV s study in chapters 3 and 4 If it is identical to one of these moment generating functions the probability distribution of U can be identi ed because of the uniqueness theorem Uniqueness Theorem Let m X t and myt denote the moment generating functions of rv s X and Y respectively If both moment generating functions exist and mXt myt for all values of t then X and Y have the same probability distribution The Ingf method is often very useful for nding the distributions fo sums of independent rv s Theorem Let Y1 Y2 Yn be independent rv s with Ingf my1tmy2t mynt respectively If U 2 Y1 Y2 Yn then mUt my1t gtlt my2 t X gtlt my t The mgf method can be used to establish some interesting and useful results about the distributions of some functions of normally distributed rv s Theorem Let Y1 Y2 7Yn be independent normally distributed rv s with m and 022 for 239 12n and let a1a2an be constants If U 2 i1 then U is a normally distributed rV With EUl 2 am i1 and van Z c1303 i1 Theorem Let Y1 Y2 Yn be independent normally distributed rv s with Em m and vow of for i 12n De ne Zi by Y2 M2 72 Zz ford 12n Then i2 i1 has a X2 distribution with n df Steps of the momentgenerating method Let U be a function of the rv s Y1Y2 Yn 1 Find the moment generating function for If mUW 2 compare mUt with other well known moment generating functions If mUt mvt for all values of t then U and V have identical distributions by uniqueness theorem Theorem ECgY Z 09ypy 0 Z 9ypy 0E9Y Theorem Let Y be a discrete rV with probability function py and 910 920 gkY be K functions of Y Then 5910 920 gkY EMUN Eig2Y 39 39 EigkY Theorem Let Y be a discrete rV with probability function py then a WY EKY m EM 2 The binomial I In this section we are concerned with experiments known as binomial experiments that exhibit the following characteristics A binomial experiment possesses the following properties 1 The experiment consists of n identical trials 2 Each trial rsults in one of two outcomes We will call one outcome a success 5 and the other a failure F 3 The probability of success on a single trial is equal to p and remains the same from trial to trial The ptrobability of a failure is equal to q 2 1 p 4 The trials are independent 5 The random variable of interest is Y the number of successes observed during the n trials Notice that the random variable of interest is the number of successes observed in the n trials A success is not necessarily good in the everyday sense of the word Example Y total number of heads When we toss 2 coins De nition A random variable Y is said to have a binomial distribution based on n trials with success probability p if and only if fory012nand0 p 1 The binomial probability distribution has many applications because the binomial experiment occurs in sampling for defectives in industrial quality control in the sampling of consumer preference or voting populations and in many other physical situations Theorem MEAN AND VARIANCE FOR THE BINOMIAL Let Y be a binomial random variable based on n trials and success probability p Then7 and Probability I Interpretation of probability Probability and Inference Notation Probabilistic Model Probability of an Event sample point method Counting sample points Conditional probability laws of probability Probability of an Event event composition method Bayes rule Interpretation I Probability is a measure of one s belief in the occurrence of a future event A prabability is a value between 0 and 1 Example if the probability of rain is 5 that means that there is a 50 chance that we will get rain If the probability of rain is 1 that means it will rain for sure If the probability is 0 then it will not rain Without doubtl The relative frequency is a meaningful measure of our belief in the occurrence of an event Example the relative frequency of a head in a long series of a coin tosses is an approximation to the probability of head but it does not provide a rigorous de nition of probability Probability and Inferencel We need a theory of probability that will pemit us to calculate the probability of observing speci ed outcomes assuming that our hypothesized model is correct Example presented in class Set Notation I Notation We use capital letters A B C to denote sets of points If the elements in the set A are 01 02 and 03 we will write A a1a2a3 A is contained in B A C B Null set q S is the universal set with all the elements under consideration A U B A union B A H B A intersection B A complement of A If A H B 2 q then A and B are disjoint or mutually exclusive Distributive laws A BUCA BUA AuBnCAuBm A DeMorgan s laws no uo Probabilistic Model I An experiment is the process by which an observation is made Examples coin and die tossing measuring the IQ score of an individual The outcomes of an experiment are called events An event which can be decomposed into other events is called a compound event An event that cannot be decomposed is called simple event Sets are collections of points Then we associate a distinct point called a sample point with each and every simple event associated with an experiment A simple event is an event that cannot be decomposed Each simple event corresponds to one and only one sample point The letter E with a subscript Will be used to denote a simple event or the corresponding sample point The sample space associated With an experiment is the set consisting of all possible sample points A sample space Will be denoted by S A discrete sample space is one that contains either a nite or a countable number of distinct sample points An event in a discrete sample space S is a collection of sample points that is any subset of S Suppose S is a sample space associated With an experiment To every event A in S A is a subset of S we assign a number PA called the probability of A so the following axioms hold 0 Axiom 1 PA Z 0 o Axiom 2 PS 1 o Axiom 3 If 141142143 form a sequence of pariWise mutually exclusive events in S that lS7 Aj 2 7E then PA1 UAZ uAg 213 i1 Stochastic Independence I We present now a formal de nition of independence of rv s De nition Let Y1 have distribution function F1y1 Y2 have distribution function F2y2 and Y1 and Y2 have joint distribution function F 31 32 Then7 Y1 and Y2 are said to be independent if and only if IKE1312 F131F2Z2 for every pair of real numbers 31 32 If Y1 and Y2 are not independent they are said to be dependent Theorem If Y1 and Y2 are discrete rV With joint probability function py1 32 and marginal probability function p1y1 and 19232 respectively then Y1 and Y2 are independent if and only if M311 32 p1y2p2y2 for all pair of real numbers 311312 Let Y1 and Y2 be jointly continuous rV with joint density fy1 32 and marginal densities f1y1 and f2y2 respectively Then Y1 and Y2 are independent if and only if flty17 32 f132f232 for all pair of real numbers 311312 Theorem Let Y1 and Y2 be jointly continuous rv with joint density fy1 32 that is positive if and only if a g 31 3 b and C 3 32 3 d for constants ab C and d and fy1 32 0 otherwise then Y1 and Y2 are independent if and only if f31Z2 gfyzlhfyzl where gy1 is a nonnegative function of y1 alone and Myg is a nonnegative function of 32 alone The key bene t of this theorem is that we do not actually need to derive the marginal densities Indeed the functions gy1 and Myg need not themselves be density functions although they will be constant multiples of the marginal densities should we go to the bother of determining the marginal densities Expected value I De nition Let gY1 Y2 Yk be a function of the discrete rv Y1 Y2 YK which have probability function py1 32 Then the expected value of 901 Y2 Yk is EEO116 Yk 2 zZzgy1ygykpyly2myk 9k 212 41 If Y1 Y2 YK are continuous rv With joint density function fy1 32 yk then EEO116 Yk 2 9y1y2ykfy17y2yk yk 342 41 13161312 05 We obtain by using the previous de nition with 991 32 11 Eiylizylf3117312d311d312 342 41 0 31 f317312dy2l dyl The quantity in brackets is called the marginal density function for Y1 and denoted as f1y1 Discrete Random Variablesl De nition of rV 211 Discrete rv 31 Probability of a discrete rV 32 The expected value 33 Binomial 34 Geometric 35 Negative binomial 36 Hypergeometric 37 Poisson 38 Moment generating functions 39 De nition of rv I Let Y a variable to be measured in an experiment Because the value of Y Will vary depending on the outcome of the experiment it is called a random variable To each point in the sample space we assign a real number denoting the value of the variable Y The value assigned to Y Will vary from one sample point to another De nition A random variable is a real value function for which the domain is a sample space Let 3 denote an observed value of the random variable Y Then PY y is the sum of the probabilities of the sample points that are assigned the value 3 Discrete rv I A random variable is used to identify numerical events that are of interest For example the event of interest in an opinion poll regarding voter preferences is Y the number of voters favoring a certain candidate or issue This random varialbe can take on only a nite number of values With nonzero probability Then Y is said to be a discrete rv De nition A random variable Y is said to be discrete if it can assume only a nite or countably in nite number of distinct values Probability Distribution I The expression Y y is read the set of all points in S39 assigned the value 1 by the random variable Y The probability that Y takes on the value 3 PY 2 y is de ned as the sum of the probabilities of all sample points in S that are assigned that value 3 We Will sometimes denote PY 2 y by The y is a function that assigns probabilities to each value 3 hence it is sometimes called the probability function for Y The probability distribution for a discrete variable Y can be represented by a formula a table or a graph Which provides for all 3 Notice that py gt 0 for all y a discrete rv assigns nonzero probabilities to only a countable number of distinct g values Theorem For any discrete probability distribution the following must be true 1 0 g py g 1 for ally 2 2y py 2 1 Where the summation is over all values of y With nonzero probability The probability distribution of a rv is a theoretical model for the empirical distribution of data associated With a real population We attempt to nd the mean and variance for a random variable Two laws of probabilityl The following two laws give the probabilities of unions and intersections of events Theorem The Multiplicative Law of Probability The probability of the intersection of two events A and B is PAB PAPBA PBPAB If A and B are independent then PAB PAPB Theorem The Additive Law of Probability The probability of the union of two events A and B is PA U B PA PB PAB If A and B are mutually exclusive events PAB 0 and PA u B PA PB Theorem If A is an event then PA 1 PUD Sometimes is easier to calculate PUD than to calculate PA In such cases it is easier to n PA by the relationship PA 2 1 PUD Moments I Generally the mean and the variance do not uniquely determine a distribution we need to study higher moments De nition If Y is a continuous rv then the kth moment about the origin is given by ukEY k12 The Kth moment about the mean or the kth central moment is given by Mk EKY Mk k12 De nition If Y is a continuous rv the the momentgenerating function of Y is given by Wt E 6 The moment generating function is said to exist if there exists a constant 6 gt 0 such that mt is nite for It 3 b Sometimes we are interested in the probability distribution of functions of rv Theorem Let Y be rV with density function y and gy be a function of Y Then the moment generating function for gY is 00 EetgY etgyfydy OO Tchebysheff s Theorem I Even if the exact distributions are unknown for rv of interest knowledge of the associated means and standard deviations permits us to deduce meaningful bounds for the probabilities of events that are often of interest Tchebyshe s Theorem Let Y be a rv With nite mean u and variance 02 Then for any k gt 0 1 Plyulltk021E 0139 1 Rig12 kmquot S E The Multinomial I A multinomial experiment is a generalization of the binomial experiment De nition A multinomial experiment possesses the following properties 1 The experiment consists of n identical trials The outcome of each trial falls into one of d classes or cells The probability that the outcome of a single trial falls into cell i is 191 i 2 1 2 k and remains the same from trial to trial Notice that P1p2quot39pk1 The trials are independent The rV s of interest are Y1 Y2 Yk Where Yi equals the number of trials for which the outcome falls into cell 239 Notice De nition Assume that 191192 pk are such that Zlepi 2 1 andpi gt 0 for i 2 12k The rv s Y1Y2 Yk are said to have a multinomial distribution with parameters n and 191192 pk if the joint probability function of Y1Y2 Yk is given by 1931732 7 n 341 yk p p 21 wk 1 k Where for each i yi 01 n and k Example of multinomial classifying people into ve income brackets results in an enumeration or count corresponding to each of ve income classes Theorem If Y1 32 Yk have a multinomial distribution with parameters n and 191192 pk then 1 EGG 71192 VYz 71192612 2 CoxYSJt npspt if 5 7E t Marginal and Conditionall De ne Y1 2 number of dots on the upper face of die 1 and Y2 is the number of dots on the upper face of die 2 For all distinct paris of values 31 32 the bivariate events Y1 31116 2 32 represented by 311312 are mutually exclusive events Then the univariate event Y1 31 is the union of bivariate events of the type Y1 31116 2 32 With the union being taken over all possible values for 32 Then 1901 2 31 19131 ENE1312 When 31 2 12 6 it is the marginal probability function for Y1 De nition 0 Let Y1 and Y2 be jointly discrete I39V With probability function py1 32 Then the marginal probability functions of Y1 and Y2 are given by 191311 EMS1312 and 19232 ENE1312 0 Let Y1 and Y2 be jointly continuous rv with joint density function f 31 32 Then the marginal density functions of Y1 and Y2 are given by mm 00 fy1yzdyz and Man 00 fan2min De nition If Y1 and Y2 are jointly discrete rv With joint probability function py1 32 and marginal probability functions p1y1 and p2y2 respectively then the conditional discrete probability function of Y1 given Y2 is pom2 PltY1 W2 92gt me 32 provided that 19232 gt 0 De nition If Y1 and Y2 are jointly continuous rv With joint density function f 31 32 then the conditional distribution function of Y1 given Y2 32 is Flt311l312gt 1001 S yllYZ 32 PltY1 Z y17Y2 Z De nition Let Y1 and Y2 be jointly continuous 11v With joint density fy1 32 and marginal densities f1y1 and f2y2 respectively For any 32 such that f2y2 gt 0 the conditional density of Y1 given Y2 2 32 is given by fy1y2 f2Z2 and for any ylsuch that f1y1 gt 0 the conditional density of Y2 given Y1 2 31 is given by f31132 fy1yzl mlyl 2 flu1 The Hypergeometric I Suppose that a population contains a nite number N of elements that possess one of two Characteristics Thus 7 of the elements might be red and b N 7 black A sample of n elements is randomly selected from the population and the random variable of interest is Y the number of red elements in the sample this random variable has What is known as the hypergeometric probability distribution A random variable Y is said to have a hypergeometric probability distribution if and only if 7 N 7 3 71 3 199 N 77 Where y is an integer 011 71 subject to the restrictions 3 g 7 and 7t y g N 7 Theorem If Y si a random variable with a hypergeometric distribution The Poisson I Suppose that we want to nd the probability distribution of Y the number of automobile accidents at a particular intersection during a time period of one week Y has a Poisson distribution Y can take values zero1 2 The Poisson often provides a good model for the probability distribution of the number Y of rare events that occur in a xed space time volume or any other dimensions Where A is the average value of Y ie the rate number of occurences per unit of time De nition A random variable Y is said to have a Poisson probability distribution if and only if A A M E6 Theorem MEAND AND VARIANCE FOR THE POISSON If Y is a random variable possessing a Poisson distribution with parameter A then and

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