1. Chapter 5 Discrete Random Variables and Probability Distributions Ósamfelld hending / Sundurleitar tilviljana breytur ©

2. Random Variables Hendingar random variable A random variable is a variable that takes on numerical values determined by the outcome of a random experiment. Hending er breyta sem tekur töluleg gildi sem ákvarðast af niðurstöðu slembinnar tilraunar.

3. Discrete Random Variables Ósamfelld hending discrete A random variable is discrete if it can take on no more than a countable number of values. Hending er ósamfelld ef hún getur einungis tekið á sig teljanlegan fjölda gilda

4. Discrete Random Variables (Examples) 1. The number of defective items in a sample of twenty items taken from a large shipment Fjöldi gallaðra hluta í úrtaki 20 hluta sem eru teknir úr stórri sendingu 2. The number of customers arriving at a check-out counter in an hour Fjöldi viðskiptavina sem kemur á innskráningu hótels á klukkutíma 3. The number of errors detected in a corporation’s accounts Fjöldi bókhaldsvillna í reikningum fyrirtækis

5. Continuous Random Variables continuous A random variable is continuous if it can take any value in an interval. Hending er samfelld ef hún getur tekið hvaða gildi sem er á ákveðnu bili.

6. Continuous Random Variables Samfelldar hendingar (Examples) 1. The income in a year for a family. Árstekjur fjölskyldu. 2.The amount of oil imported into the U.S. in a particular month. Útflutt magn af fiski á einu ári. 3.The change in the price of a share of IBM common stock in a month. Verðbreyting hlutabréfs. 4.The time that elapses between the installation of a new computer and its failure. Tími sem líður frá uppsetningu tölvu þar til hún bilar.

7. Discrete Probability Distributions Líkindadreifing fyrir ósamfellda hendingu probability distribution function (DPF), The probability distribution function (DPF), P(x), of a discrete random variable expresses the probability that X takes the value x, as a function of x. That is Líkindadreifingarfall, P(x), ósamfelldrar hendingar gefur til kynna líkurnar á að hending X taki gildið x, sem fall af x. Það er :

8. Discrete Probability Distributions Líkindadreifing fyrir ósamfellda hendingu (Example 5.1) Graph the probability distribution function for the roll of a single six-sided die. Líkindadreifing fyrir tening 123456 1/6 P(x) x Figure 5.1

9. Required Properties of Probability Distribution Functions of Discrete Random Variables Eiginleikar líkindadreifingar ósamfelldrar hendingar. Let X be a discrete random variable with probability distribution function, P( x ). Then i.P(x)  0 for any value of x ii.The individual probabilities sum to 1; that is Látum X vera ósamfellda hendingu með líkindadreifingarfall, P(x). Þá i.P(x)  0 fyrir sérhvert gildi á x ii.Summa á líkum allra atburða; þ.e. Where the notation indicates summation over all possible values x. Þar sem samlagningin er fyrir öll möguleg gildi á x.

10. Cumulative Probability Function Uppsafnað líkindafall cumulative probability function, The cumulative probability function, F(x 0 ), of a random variable X expresses the probability that X does not exceed the value x 0, as a function of x 0. That is Uppsafnað líkindafall, F(x0), hendingar X gefur til kynna líkurnar á því að X sé minna en x0, sem fall af x0. Þ.e. Where the function is evaluated at all values x 0 Þar sem fallið tekur gildið við öll möguleg x 0

11. Derived Relationship Between Probability Function and Cumulative Probability Function Tengslin milli líkindadreifingarfall og uppsafnaðs líkindafalls Let X be a random variable with probability function P(x) and cumulative probability function F(x 0 ). Then it can be shown that Látum X vera hendingu með líkindafall P(x) og uppsafnað líkindafall F(x 0 ). Þá er hægt að sýna að Where the notation implies that summation is over all possible values x that are less than or equal to x 0. Þar sem samlagning er fyrir öll möguleg gildi á x sem eru lægri en eða jöfn x 0.

12. Derived Properties of Cumulative Probability Functions for Discrete Random Variables Eiginleikar uppsafnaðs líkindafalls fyrir ósamfelldar hendingar. Let X be a discrete random variable with a cumulative probability function, F( x 0 ). Then we can show that i.0  F( x 0 )  1 for every number x 0 ii.If x 0 and x 1 are two numbers with x 0 < x 1, then F(x 0 )  F(x 1 ) Látum X vera ósamfellda hendingu með uppsafnað líkindafall, F( x 0 ). Þá má sýna að: i.0  F( x 0 )  1 fyrir sérhvert gildi á x 0 ii.Ef x 0 og x 1 eru tvær tölur með x 0 < x 1, þá gildir F(x 0 )  F(x 1 )

13. Expected Value Vontigildi /Vongildi expected value, E(X), The expected value, E(X), of a discrete random variable X is defined Væntanlegt gildi, E(X), ósamfelldrar hendingar X er skilgreint sem: Where the notation indicates that summation extends over all possible values x. mean  x The expected value of a random variable is called its mean and is denoted  x. Þar sem samlagning nær yfir öll möguleg gildi á x.  x Vongildi hendingar kallast meðaltal hendingar og er táknað með  x.

14. Expected Value: Functions of Random Variables Vongildi: Fall af hendingu Let X be a discrete random variable with probability function P(x) and let g(X) be some function of X. Then the expected value, E[g(X)], of that function is defined as Látum X vera ósamfellda hendingu með líkindafall P(x) og látum g(X) vera fall af X. Þá er vongildi þess falls, E[g(X)], skilgreint sem:

15. Variance and Standard Deviation Dreifni og staðalfrávik variance Let X be a discrete random variable. The expectation of the squared discrepancies about the mean, (X -  ) 2, is called the variance, denoted  2 x and is given by dreifni variance Látum X vera ósamfellda hendingu. Vongildi frávika í öðru veldi frá meðaltali hendingarinnar, (X -  ) 2, kallast dreifni variance, táknað með  2 x og er reiknað sem: standard deviation The standard deviation,  x, is the positive square root of the variance. Staðalfrávikið,  x, er jákvæð kvaðratrót dreifninnar.

16. Variance Dreifni (Alternative Formula) variance The variance of a discrete random variable X can be expressed as

17. Expected Value and Variance for Discrete Random Variable Using Microsoft Excel Vongildi og dreifni fyrir ósamfellda hendingu með notkun Excel (Figure 5.4) Expected Value = 1.95Variance = 1.9475

18. Summary of Properties for Linear Function of a Random Variable Eiginleikar línulegs falls af hendingu meanvariance Let X be a random variable with mean  x, and variance  2 x ; and let a and b be any constant fixed numbers. Define the random variable Y = a + b X. Then, the mean and variance of Y are Látum X vera hendingu með meðaltal  x, og dreifni  2 x ; og látum a og b vera fasta. Skilgreinum nú hendinguna Y = a + b X. Þá eru meðaltal og dreifni Y and standard deviation of Y so that the standard deviation of Y is Þannig verður staðalfrávik Y

19. Summary Results for the Mean and Variance of Special Linear Functions Nokkrar reglur um vongildi og dreifni fyrir sérstök línuleg föll a) Let b = 0 in the linear function, W = a + bX. Then W = a (for any constant a). Látum b = 0 í línulega fallinu, W = a + bX. Þá er W = a If a random variable always takes the value a, it will have a mean a and a variance 0. Ef hending tekur alltaf gildið a mun hún hafa það meðaltal og dreifni 0. b)Let a = 0 in the linear function, W = a + bX. Then W = bX. Látum a = 0 í línulega fallinu, W = a + bX. Þá er W = bX.

20. Mean and Variance of Z Meðaltal og dreifni Z Let a = -  X /  X and b = 1/  X in the linear function Z = a + bX. Then, so that and

21. Bernoulli Distribution Bernoulli dreifing Bernoulli distribution A Bernoulli distribution arises from a random experiment which can give rise to just two possible outcomes. These outcomes are usually labeled as either “success” or “failure.” If  denotes the probability of a success and the probability of a failure is (1 -  ), the the Bernoulli probability function is Bernoulli dreifing verður til í slembinni tilraun sem einungis getur leitt af sér tvær niðurstöður. Þessar niðurstöður eru oft merktar sem tilraun “heppnist” eða “misheppnist” Ef  táknar líkur þess að tilraun heppnst og þá eru líkurnar á því að tilraun misheppnist: (1 -  ), Bernoulli líkindafallið verður því

22. Mean and Variance of a Bernoulli Random Variable mean The mean is: variance And the variance is:

23. Sequences of x Successes in n Trials Fjöldi tilrauna x sem heppnast í n tilraunum number of sequences with x successes in n independent trials The number of sequences with x successes in n independent trials is: Where n! = n x (n – 1) x (n – 2) x... x 1 and 0! = 1.

24. Binomial Distribution Tvíliðunardreifingin binomial distribution Suppose that a random experiment can result in two possible mutually exclusive and collectively exhaustive outcomes, “success” and “failure,” and that  is the probability of a success resulting in a single trial. If n independent trials are carried out, the distribution of the resulting number of successes “x” is called the binomial distribution. Its probability distribution function for the binomial random variable X = x is: Hugsum okkur að slembin tilraun getu leitt til tveggja mögulegra niðurstaðna sem eru sundurlægir atburðir og samanlagt tæmandi. Köllum atburðina “heppnast” “misheppnast” og að  séu líkurnar á því að þegar tilraun er framkvæmd einu sinni heppnist hún. Ef n óháðar tilraunir eru framkvæmdar þá er hending fjölda tilrauna þar sem atburðurinn heppnaðist táknuð með “X” og dreifing hendingarinnar er kölluð Tvíliðunardreifing. Líkindadreifingarfall P( x successes in n independent trials)= for x = 0, 1, 2..., n

25. Mean and Variance of a Binomial Probability Distribution Vongildi og dreifni tvíliðunardreifingar mean Let X be the number of successes in n independent trials, each with probability of success . The x follows a binomial distribution with mean, Látum X vera fjölda tilrauna þar sem atburður heppnast í n óháðum tilraunum, þar sem sérhver tilraun gefur líkur  á að atburður heppnist. Þá mun X hafa tvíliðunardreifing með meðfylgjandi meðaltali variance and variance,

26. Binomial Probabilities - An Example – (Example 5.7) An insurance broker, Shirley Ferguson, has five contracts, and she believes that for each contract, the probability of making a sale is 0.40. What is the probability that she makes at most one sale? P(at most one sale) = P(X  1) = P(X = 0) + P(X = 1) = 0.078 + 0.259 = 0.337

27. Binomial Probabilities, n = 100,  =0.40 (Figure 5.10)

28. Hypergeometric Distribution Happadreifingin hypergeometric distribution Suppose that a random sample of n objects is chosen from a group of N objects, S of which are successes. The distribution of the number of X successes in the sample is called the hypergeometric distribution. Its probability function is: Hugsum okkur að slembið úrtak n hluta er valið úr hópi N hluta, S þeirra teljast “heppnaðir”. Dreifing fjölda heppnaðara X í úrtakinu er kölluð Happadreifingin. Líkindafall þessarar dreifingar er: Where x can take integer values ranging from the larger of 0 and [n-(N-S)] to the smaller of n and S.

29. Poisson Probability Distribution Poisson dreifing Poisson probability distribution Assume that an interval is divided into a very large number of subintervals so that the probability of the occurrence of an event in any subinterval is very small. The assumptions of a Poisson probability distribution are: 1)The probability of an occurrence of an event is constant for all subintervals. 2)There can be no more than one occurrence in each subinterval. 3)Occurrences are independent; that is, the number of occurrences in any non-overlapping intervals in independent of one another. Gerum ráð fyrir að tímabili sé skipt í mjög marga parta þannig að líkurnar á að atburður eigi sér stað í sérhverjum parti eru mjög lágar. Forsendur Poisson líkindadreifingar eru eftirfarandi: 1)Líkurnar á atburði í serhverjum parti eru eins fyrir alla partana. 2)Það getur aldrei orðið meira en einn atburður í hverjum tímaparti. 3)Atburðirnir eru óháðir.

30. Poisson Probability Distribution The random variable X is said to follow the Poisson probability distribution if it has the probability function: Hending X er sögð fylgja Poisson líkindadreifingu ef hún hefur eftirfarandi líkindafall: where P(x) = the probability of x successes over a given period of time or space, given = the expected number of successes per time or space unit; > 0 e = 2.71828 (the base for natural logarithms) mean and variance of the Poisson probability distribution are The mean and variance of the Poisson probability distribution are :

31. Partial Poisson Probabilities for = 0.03 Obtained Using Microsoft Excel PHStat (Figure 5.14)

32. Poisson Approximation to the Binomial Distribution Poisson nálgun fyrir tvíliðunardreifinguna Let X be the number of successes resulting from n independent trials, each with a probability of success, . The distribution of the number of successes X is binomial, with mean n . If the number of trials n is large and n  is of only moderate size (preferably n   7), this distribution can be approximated by the Poisson distribution with = n . The probability function of the approximating distribution is then: Látum X vera fjölda heppnaðra atburða úr n óháðum tilraunum þar sem sérhver tilraun hefur líkurnar  á því að heppnast. Dreifing fjölda heppnaðra atburða X hefur tvíliðunardreifingu með vongildi n . Ef fjöldi tilrauna n er stór stærð og n  er af hóflegri stærð helst n   7, þá má nálga þessa dreifingu með Poisson dreifingunni með = n . Líkindafall þessarar nálgunardreifing verður þá:

33. Joint Probability Functions Sameiginleg líkindaföll joint probability function Let X and Y be a pair of discrete random variables. Their joint probability function expresses the probability that X takes the specific value x and simultaneously Y takes the value y, as a function of x and y. The notation used is P(x, y) so, Látum X og Y vera hendingar. Sameiginlegt líkindafall þeirra gefur til kynna líkurnar á því að hendingin X taki á sig ákveðið gildi x og að á sama tíma taki hendingin Y á sig gildið y, líkurnar eru fall af x og y. Þetta er táknað með P(x, y) þannig að,

34. Joint Probability Functions Sameiginleg líkindaföll marginal probability function Let X and Y be a pair of jointly distributed random variables. In this context the probability function of the random variable X is called its marginal probability function and is obtained by summing the joint probabilities over all possible values; that is, Látum X og Y vera par af sameiginlega dreifðum hendingum. Í þessu samhengi er líkindafall X kallað jaðarlíkindafall og er fengið með því að leggja saman sameiginlega líkindafallið yfir öll möguleg gildi á y fyrir hvert gildi á x, marginal probability function Similarly, the marginal probability function of the random variable Y is Á sama hátt er jaðarlíkindafall Y fengið:

35. Properties of Joint Probability Functions 4 Let X and Y be discrete random variables with joint probability function P(x,y). Then 1.P(x,y)  0 for any pair of values x and y 2.The sum of the joint probabilities P(x, y) over all possible values must be 1.

36. Conditional Probability Functions Skilyrt líkindaföll conditional probability function Let X and Y be a pair of jointly distributed discrete random variables. The conditional probability function of the random variable Y, given that the random variable X takes the value x, expresses the probability that Y takes the value y, as a function of y, when the value x is specified for X. This is denoted P(y|x), and so by the definition of conditional probability: Skilyrt líkindafall Látum X og Y vera par sameiginlega dreifða hendinga. Skilyrt líkindafall hendingarinnar Y að gefnu því að hendingin X taki gildið x, gefur til kynna líkurnar á því að að Y taki gildið y, sem fall af y, þegar gildið x er tiltekið fyrir X. Þetta er táknað með P(y|x), þar af leiðandi samkvæmt skilgreiningu á skilyrtum líkindum: conditional probability function Similarly, the conditional probability function of X, given Y = y is:

37. Independence of Jointly Distributed Random Variables Óháðar sameiginlega dreifðar hendingar independent The jointly distributed random variables X and Y are said to be independent if and only if their joint probability function is the product of their marginal probability functions, that is, if and only if Sameiginlega dreifðar hendingar X og Y eru sagðar óháðar ef og aðeins ef sameiginlega líkindafall er margfeldi jaðarlíkindafalla X og Y. Þ.e. ef og aðeins ef. And k random variables are independent if and only if

38. Expected Value Function of Jointly Distributed Random Variables Vongildi sameiginlega dreifðra hendinga expectation of any function g(x, y) Let X and Y be a pair of discrete random variables with joint probability function P(x, y). The expectation of any function g(x, y) of these random variables is defined as: Látum X og Y vera pört ósamfelldra hendinga með sameiginlegt líkindadreifingafall P(x, y). Vongildi sérhvers falls g(x, y) Vongildi sérhvers falls g(x, y) þessara hendinga er skilgreint sem:

39. Stock Returns, Marginal Probability, Mean, Variance (Example 5.16) Y Return X Return 0%5%10%15% 0%0.0625 5%0.0625 10%0.0625 15%0.0625 Table 5.6

40. Covariance Samdreifni covariance Let X be a random variable with mean  X, and let Y be a random variable with mean,  Y. The expected value of (X -  X )(Y -  Y ) is called the covariance between X and Y, denoted Cov(X, Y). For discrete random variables Samdreifni Látum X vera hendingu með vongildi  X, og látum Y vera hendingu með vongildi,  Y. Vongildi margfeldis (X -  X )(Y -  Y ) er kallað Samdreifni milli X og Y, táknað með Cov(X, Y). Fyrir ósamfelldar hendingar: An equivalent expression is Jafngild framsetning:

41. Correlation Fylgni Let X and Y be jointly distributed random variables. The correlation between X and Y is: Látum X og Y vera sameiginlega dreifðar hendingar Fylgni milli X og Y:

42. Covariance and Statistical Independence statistically independent If two random variables are statistically independent, the covariance between them is 0. However, the converse is not necessarily true. Ef tvær hendingar eru tölfræðilega óháðar þá er samdreifni milli þeirra 0. Ef samdreifni er 0 gefur það hins vegar ekki endilega til kynna að þær séu óháðar.

43. Portfolio Analysis Greining safns The random variable X is the price for stock A and the random variable Y is the price for stock B. The market value, W, for the portfolio is given by the linear function, Hending X er verð bréfs A og hending Y er verð bréfs B. Markaðsvirði W fyrir safnið er gefið af línulega fallinu. Where, a, is the number of shares of stock A and, b, is the number of shares of stock B. Þar sem a er fjöldi bréfa A og b er fjöldi bréfa B.

44. Portfolio Analysis Greining safns mean value for W The mean value for W is, variance for W The variance for W is, or using the correlation, skilgreiningu fylgni

45. Key Words 4Bernoulli Random Variable, Mean and Variance 4Binomial Distribution 4Conditional Probability Function 4Continuous Random Variable 4Correlation 4Covariance 4Cumulative Probability Function 4 Differences of Random Variables 4 Discrete Random Variable 4 Expected Value 4 Expected Value: Functions of Random Variables 4 Expected Value: Function of Jointly Distributed Random Variable 4 Hypergeometric Distribution 4 Independence of Jointly Distributed Random Variables

46. Key Words (continued) 4Joint Probability Function 4Marginal Probability Function 4Mean of Binomial Distribution 4Mean: Functions of Random Variables 4Poisson Approximation to the Binomial Distribution 4Poisson Distribution 4Portfolio Analysis 4 Portfolio, Market Value 4 Probability Distribution Function 4 Properties: Cumulative Probability Functions 4 Properties: Joint Probability Functions 4 Properties: Probability Distribution Functions 4 Random Variable

47. Key Words (continued) 4Relationships: Probability Function and Cumulative Probability Function 4Standard Deviation: Discrete Random Variable 4Sums of Random Variables 4Variance: Binomial Distribution 4 Variance: Discrete Random Variable 4 Variance: Discrete Random Variable (Alternative Formula) 4 Variance: Functions of Random Variables