Fundamentals of Mathematical Statistics - uniza.sk

1 Fundamentals of Mathematical StatisticsPavol OR ANSK 2 Contents1 Probability Random event .. Algebraic operations and programs with events .. Classical de nition of probability .. Kolmogorov de nition of probability .. Probability of the uni cation of random events .. Probality of the opposite event .. Conditional probability .. Intersection probability of random phenomena .. Full probability formula .. Bayesov vzorec .. Bernoulliho vzorec .. 232 Random Discrete probability distribution .. Distribution function of random variable .. Density distribution.

2 Basic features of the density distribution.. Numerical characteristics of random variable .. Mean value .. Variance (dispersion) and standard deviation .. 333 Signi cant continuous distribution of random Normal distribution (Laplace-Gaussovo distribution) .. Standard normal distribution .. The relationship betweenF(x)and (x).. 384 Descriptive statistics415 Estimates of Point estimate .. Interval estimation of parameters .. (1 )%bilated con dence interval for mean value (1 )%bilateral con dence interval for dispersion 25334 CONTENTS6 Testing statistical Parametric testing an single le.

3 Testing parameter if the set is small(n 30).. Testing parameter if the set is large(n >30).. Testing parameter .. Comparing two les .. Testing equality of means of two fundamental les, whenthe les are large(n >30), and if 1, 2are known .. Testing equality of means of two fundamental les, whenthe les are small(n <30), and if 1, 2are known .. Testing equality of means of two fundamental les, whenthe les are large(n >30), and if 1, 2are unknown .. Testing equality of means of two fundamental les, whenthe les are small(n <30), and if 1, 2are unknown .. 657 Correlation Coe cient covariance.

4 Correlation coe cient .. Coe cient of determination .. 708 Paired linear Regression line .. Estimation of the parameters 0and 1.. 789 Attachments81 PrefaceThis text was based on the inherent requirements of my students on the subjectof Statistics Faculty of Management, University of Pre ov, an overview of acomprehensive publication of purely on Statistics , contained in their basic existing text was either too large, and thus a deterrent to the reader, or,conversely, some publications contained only secondary curriculum. Demandson university students has recently changed considerably, therefore was also agap in the literature, which lacks books covering a kind of intermediate stage ofsecondary literature and literature explicitly university type, ie.

5 Klad ce booksto the reader requirements less stringent than in the past. This gap, I tried the"quick x" patch overwrite my lectures from the course Statistics in acceptableform. I have text in addition to enriching addressed but not resolved by theexamples that I drew from its own resources respectively. I took them from thebook [1].1 This text does not replace any, in my opinion, excellent publications by otherauthors, of which I once again mention Chajdiaks [1]. But in some way tryingto bring modern students who are trying, let s face it, as far as possible to savemathematics. It is strictly text reader for inexpensive, lightweight for deeperanalysis of , I would like to thank my colleague and good friend Dr.

6 , Ing. J n RY-B RIK, PhD., For their support and valuable advice of an experienced teacher,without whom this text would not arise. Also thank my students for c n, May 30, 2009 Pavol OR ANSK 1No results or arguments and examples used in them are not based on facts, and thereforethere is no real pre guration them, which could be an eventual usurper can look to the translationThis text was created, for the purpose of teaching foreign students, as a trans-lation of teaching material originally intended for teaching at the University ofPre ov. Transfer and reducing the learning material we have created a textthat covers the issue of Statistics needed to manage the course of numerical anddiscrete chapters present the minimum necessary for basic understandingof statistical methods.

7 The rst chapter de ned the very concept of probability,we deal with here its basic the second and third chapter describes some basic probability the fourth chapter , the Fundamentals of descriptive Statistics , the neces-sary basis for the data fth chapter deals with the estimation of sixth chapter is devoted to testing statistical hypotheses about the pa-rameters and eighth chapter discusses the issue of linear dependency c n, January 2011 Pavol OR ANSK 78 CONTENTSC hapter 1 Probability theoryProbability theory is a kind of basis for Statistics . Is an integral part thereof. Fora deeper understanding of Statistics is necessary to have knowledge of at leastthis chapter .

8 Probability theory describes random events and probability of theoccurrence. Statistics speci c model empirical events, and statistical methodsto describe these events have a fundamental right in probability Random eventDe nition 1 Random attempt (plot)is an attempt, the outcome is notclearly de ned conditions, which were carried out. We have in mind attempts(even if the term ts better story), whose outcome can not be determined example: Coin toss or dice, the number of dead after a moderate earth-quake in China, the possibility of TB infection using trains Slovak railways, nition 2 Random eventis true that the result of an accidental experi-ment.

9 In other words, the outcome of the example: "When throwing a coin falls symbol character", "The roll ofdice you get the number ve", "When a moderate earthquake in China killed3624 people" or "While driving Slovak railways would be infected with TB everyone hundred fty-nine billionth (wonderful word) the passengers .Random events will be marked in capitals, "Hugo Chaves wins the presidential election in Venezuela."In our view, are also interesting attempts, which will be studied event, theincreasing number of experiments show some stability. Which means that therelative frequency of the occurrence of accidental eventAhas an increasing910 chapter 1.

10 PROBABILITY THEORY number of trials tends to be constant. Where relative abundance is expressedas the ratio of "the occurrence of the eventA" to "of all nothing other thannAn konst:, forn 1;nAindicates how many times there was a eventA,nindicates the number of times an attempt was call this constant probability of accidental eventA. (For the exampleabove, this probability is equal to the probability of the same event, as you saylater.)For illustration, consider the dice with six sides and a random eventAthat"throwing the dice when you get the number ve".If we started to throw the dice and the results we would be registered, aftera su cient number of throws we would have noticed that the number of fallingnumbers ve and the number of roll the dice in a ratio of about 1 to 6th Thisresult would probably not surprising, since almost everyone also expects thatthe probability of "throwing the dice when you get the number ve" is16=0:166 67$16:7%.