Chapter 5Chapter 5

Discrete Random Discrete Random Variables and Variables and

Probability Probability DistributionsDistributions

Ósamfelld hending /Ósamfelld hending /

Sundurleitar tilviljana breyturSundurleitar tilviljana breytur

©

Random VariablesRandom Variables HendingarHendingar

A random variablerandom 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.

Discrete Random VariablesDiscrete Random VariablesÓsamfelld hendingÓsamfelld hending

A random variable is discrete 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

Discrete Random VariablesDiscrete Random Variables(Examples)(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

Continuous Random Continuous Random VariablesVariables

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

Continuous Random Continuous Random VariablesVariables

Samfelldar hendingarSamfelldar hendingar(Examples)(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.

Discrete Probability DistributionsDiscrete Probability Distributions Líkindadreifing fyrir ósamfellda hendinguLíkindadreifing fyrir ósamfellda hendingu

The probability distribution function probability distribution function (DPF),(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 : . of valuesallfor ),()( xxXPxP

Discrete Probability DistributionsDiscrete Probability DistributionsLíkindadreifing fyrir ósamfellda hendinguLíkindadreifing fyrir ósamfellda hendingu

(Example 5.1)(Example 5.1)

Graph the probability distribution function for the roll of a single six-sided die.

Líkindadreifing fyrir tening

1 2 3 4 5 6

1/6

P(x)

x

Figure 5.1Figure 5.1

Required Properties of Probability Required Properties of Probability Distribution Functions of Discrete Random Distribution Functions of Discrete Random

Variables Variables Eiginleikar líkindadreifingar Eiginleikar líkindadreifingar ósamfelldrar hendingar.ósamfelldrar hendingar.

Let X be a discrete random variable with probability distribution function, P(x). Then

i. P(x) 0 for any value of xii. 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 xii. 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.

x

xP 1)(

Cumulative Probability Cumulative Probability FunctionFunction

Uppsafnað líkindafallUppsafnað líkindafall

The cumulative probability cumulative probability function,function, F(x0), of a random variable X expresses the probability that X does not exceed the value x0, as a function of x0. 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 x0

Þar sem fallið tekur gildið við öll möguleg x0

)()( 00 xXPxF

Derived Relationship Between Probability Derived Relationship Between Probability Function and Cumulative Probability Function and Cumulative Probability

FunctionFunctionTengslin milli líkindadreifingarfall og uppsafnaðs Tengslin milli líkindadreifingarfall og uppsafnaðs

líkindafallslíkindafalls

Let X be a random variable with probability function P(x) and cumulative probability function F(x0). Then it can be shown thatLátum X vera hendingu með líkindafall P(x) og uppsafnað líkindafall F(x0). 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 x0.Þar sem samlagning er fyrir öll möguleg gildi á x sem eru lægri en eða jöfn x0.

0

)()( 0xx

XPxF

Derived Properties of Cumulative Probability Derived Properties of Cumulative Probability Functions for Discrete Random VariablesFunctions for Discrete Random Variables

Eiginleikar uppsafnaðs líkindafalls fyrir ósamfelldar hendingar. Eiginleikar uppsafnaðs líkindafalls fyrir ósamfelldar hendingar.

Let X be a discrete random variable with a cumulative probability function, F(x0). Then we can show that

i. 0 F(x0) 1 for every number x0

ii. If x0 and x1 are two numbers with x0 < x1, then F(x0) F(x1)

Látum X vera ósamfellda hendingu með uppsafnað líkindafall, F(x0). Þá má sýna að:

i. 0 F(x0) 1 fyrir sérhvert gildi á x0

ii. Ef x0 og x1 eru tvær tölur með x0 < x1, þá gildir F(x0) F(x1)

Expected Value Expected Value VoVontigildi /Vongildintigildi /Vongildi The expected value, E(X),expected value, E(X), of a discrete random variable X is definedVæntanlegt gildi, E(X), ósamfelldrar hendingar X er skilgreint sem:

Where the notation indicates that summation extends over all possible values x.The expected value of a random variable is called its meanmean and is denoted xx.Þar sem samlagning nær yfir öll möguleg gildi á x.Vongildi hendingar kallast meðaltal hendingar og er táknað með xx.

x

xxPXE )()(

Expected Value: Functions Expected Value: Functions of Random Variables of Random Variables

Vongildi: Fall af hendinguVongildi: 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 asLátum 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:

x

xPxgXgE )()()]([

Variance and Standard DeviationVariance and Standard Deviation Dreifni og staðalfrávikDreifni og staðalfrávik

Let X be a discrete random variable. The expectation of the squared discrepancies about the mean, (X - )2, is called the variancevariance, denoted 2

x and is given by

Látum X vera ósamfellda hendingu. Vongildi frávika í öðru veldi frá meðaltali hendingarinnar, (X - )2, kallast dreifni dreifni variancevariance, táknað með 2

x og er reiknað sem:

The standard deviationstandard deviation, x , is the positive square root of the variance.Staðalfrávikið, x , er jákvæð kvaðratrót dreifninnar.

x

xxx xPxXE )()()( 222

Variance Variance Dreifni Dreifni (Alternative Formula) (Alternative Formula)

The variancevariance of a discrete random variable X can be expressed as

22

222

)(

)(

xx

xx

xPx

XE

Expected Value and Variance for Expected Value and Variance for Discrete Random Variable Using Discrete Random Variable Using

Microsoft ExcelMicrosoft ExcelVongildi og dreifni fyrir ósamfellda Vongildi og dreifni fyrir ósamfellda

hendingu með notkun Excelhendingu með notkun Excel (Figure 5.4)(Figure 5.4)

Sales P(x) Mean Variance0 0.15 0 0.5703751 0.3 0.3 0.270752 0.2 0.4 0.00053 0.2 0.6 0.22054 0.1 0.4 0.420255 0.05 0.25 0.465125

1.95 1.9475

Expected Value = 1.95 Variance = 1.9475

Summary of Properties for Linear Function of a Summary of Properties for Linear Function of a Random Variable Random Variable Eiginleikar línulegs falls af Eiginleikar línulegs falls af

hendinguhendinguLet 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 + bX. Then, the meanmean and variancevariance 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 + bX. Þá eru meðaltal og dreifni Y

and

so that the standard deviation of Ystandard deviation of Y is Þannig verður staðalfrávik Y

XY babXaE )(

XY bbXaVar 222 )(

XY b

Summary Results for the Mean and Variance Summary Results for the Mean and Variance of Special Linear Functionsof Special Linear Functions

Nokkrar reglur um vongildi og dreifni fyrir sérstök Nokkrar reglur um vongildi og dreifni fyrir sérstök línuleg fölllínuleg 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.

0)()( aVarandaaE

22)()( XX baVarandbbXE

Mean and Variance of ZMean and Variance of Z Meðaltal og dreifni ZMeðaltal og dreifni Z

Let a = -X/X and b = 1/ X in the linear function Z = a + bX. Then,

so that

and

X

XXbXaZ

01

X

XX

X

X

XXE

11 2

2

X

XX

XXVar

Bernoulli DistributionBernoulli DistributionBernoulli dreifingBernoulli dreifing

A Bernoulli distributionBernoulli 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

)1()1()0( PandP

Mean and Variance of a Mean and Variance of a Bernoulli Random VariableBernoulli Random Variable

The meanmean is:

And the variancevariance is:

)1()1)(0()()( xPxXEX

X

)1()1()1()0(

)()(])[(

22

222

X

XXX xPxXE

Sequences of Sequences of xx Successes in Successes in nn Trials Trials Fjöldi tilrauna x sem heppnast í n tilraunumFjöldi tilrauna x sem heppnast í n tilraunum

The number of sequences with x successes in n number of sequences with x successes in n independent trialsindependent trials is:

Where n! = n x (n – 1) x (n – 2) x . . . x 1 and 0! = 1.

)!(!

!

xnx

nC n

x

time. samethe at occur can them of two no since

exclusive,mutually are sequencesC These nx

Binomial Distribution Binomial Distribution TvíliðunardreifinginTvíliðunardreifingin 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 binomial distributiondistribution. 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

)()1()!(!

!)( xnx

xnx

nxP

Mean and Variance of a Binomial Probability Mean and Variance of a Binomial Probability DistributionDistribution

Vongildi og dreifni tvíliðunardreifingarVongildi og dreifni tvíliðunardreifingar

Let X be the number of successes in n independent trials, each with probability of success . The x follows a binomial distribution with meanmean,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

and variancevariance,

nXEX )(

)1(])[( 22 nXEX

Binomial ProbabilitiesBinomial Probabilities- An Example –- An Example –

(Example 5.7)(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

259.0)6.0()4.0(1!4!

5! P(1) sale) P(1

0.078 )6.0()4.0(0!5!

5! P(0) sales) P(no

41

50

Binomial Probabilities, n = 100, Binomial Probabilities, n = 100, =0.40=0.40

(Figure 5.10)(Figure 5.10)

Sample size 100Probability of success 0.4Mean 40Variance 24Standard deviation 4.898979

Binomial Probabilities TableX P(X) P(<=X) P(<X) P(>X) P(>=X)

36 0.059141 0.238611 0.179469 0.761389 0.82053137 0.068199 0.30681 0.238611 0.69319 0.76138938 0.075378 0.382188 0.30681 0.617812 0.6931939 0.079888 0.462075 0.382188 0.537925 0.61781240 0.081219 0.543294 0.462075 0.456706 0.53792541 0.079238 0.622533 0.543294 0.377467 0.45670642 0.074207 0.69674 0.622533 0.30326 0.37746743 0.066729 0.763469 0.69674 0.236531 0.30326

Hypergeometric Distribution Hypergeometric Distribution HappadreifinginHappadreifingin

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 distributionhypergeometric 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.

)!(!!

)!()!()!(

)!(!!

)(

nNnN

xnSNxnSN

xSxS

C

CCxP

Nn

SNxn

Sx

Poisson Probability Distribution Poisson Probability Distribution Poisson Poisson dreifingdreifing

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 distributionPoisson 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.

Poisson Probability DistributionPoisson Probability DistributionThe 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:

whereP(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; > 0e = 2.71828 (the base for natural

logarithms)The mean and variance of the Poisson mean and variance of the Poisson probability distribution areprobability distribution are:

1,2,... 0,xfor,!

)(

x

exP

x

])[()( 22 XEandXE xx

Partial Poisson Probabilities for Partial Poisson Probabilities for = = 0.03 Obtained Using Microsoft 0.03 Obtained Using Microsoft

Excel Excel PPHStatHStat(Figure 5.14)(Figure 5.14)

Poisson Probabilities TableX P(X) P(<=X) P(<X) P(>X) P(>=X)0 0.970446 0.970446 0.000000 0.029554 1.0000001 0.029113 0.999559 0.970446 0.000441 0.0295542 0.000437 0.999996 0.999559 0.000004 0.0004413 0.000004 1.000000 0.999996 0.000000 0.0000044 0.000000 1.000000 1.000000 0.000000 0.000000

Poisson Approximation to the Binomial Poisson Approximation to the Binomial Distribution Distribution Poisson nálgun fyrir Poisson nálgun fyrir

tvíliðunardreifingunatvíliðunardreifingunaLet 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:

1,2,... 0,xfor,!

)()(

x

nexP

xn

Joint Probability Functions Joint Probability Functions Sameiginleg líkindaföllSameiginleg líkindaföll

Let X and Y be a pair of discrete random variables. Their joint probability functionjoint 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,

)(),( yYxXPyxP

Joint Probability Functions Joint Probability Functions Sameiginleg líkindaföllSameiginleg líkindaföllLet 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 functionmarginal 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,

Similarly, the marginal probability functionmarginal probability function of the random variable Y is Á sama hátt er jaðarlíkindafall Y fengið:

y

yxPxP ),()(

x

yxPyP ),()(

Properties of Joint Properties of Joint Probability FunctionsProbability Functions

Let X and Y be discrete random variables with joint probability function P(x,y). Then

1.1. P(x,y) P(x,y) 0 for any pair of values x and y 0 for any pair of values x and y

2.2. The sum of the joint probabilities P(x, The sum of the joint probabilities P(x, y) over all possible values must be 1.y) over all possible values must be 1.

Conditional Probability Functions Conditional Probability Functions Skilyrt líkindaföllSkilyrt líkindaföll

Let X and Y be a pair of jointly distributed discrete random variables. The conditional probability functionconditional 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: Látum X og Y vera par sameiginlega dreifða hendinga. Skilyrt Skilyrt líkindafalllíkindafall 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:

Similarly, the conditional probability functionconditional probability function of X, given Y = y is:

)(

),()|(

xP

yxPxyP

)(

),()|(

yP

yxPyxP

Independence of Jointly Distributed Random Independence of Jointly Distributed Random Variables Variables Óháðar sameiginlega dreifðar hendingarÓháðar sameiginlega dreifðar hendingar

The jointly distributed random variables X and Y are said to be independentindependent 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

y. and x valuesof pairs possible allfor )()(),( yPxPyxP

)()()(),,,( 2121 kk xPxPxPxxxP

Expected Value Function of Jointly Distributed Random Expected Value Function of Jointly Distributed Random VariablesVariables

Vongildi sameiginlega dreifðra hendinga Vongildi sameiginlega dreifðra hendinga

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)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:

x y

yxPyxgYXgE ),(),()],([

Stock Returns, Marginal Stock Returns, Marginal Probability, Mean, VarianceProbability, Mean, Variance

(Example 5.16)(Example 5.16)

Y Return

X Retur

n

0% 5% 10% 15%

0% 0.0625

0.0625

0.0625

0.0625

5% 0.0625

0.0625

0.0625

0.0625

10% 0.0625

0.0625

0.0625

0.0625

15% 0.0625

0.0625

0.0625

0.0625

Table 5.6

Covariance Covariance SamdreifniSamdreifniLet 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 covariance between X and Y, denoted Cov(X, Y).For discrete random variablesLátum X vera hendingu með vongildi X , og látum Y vera hendingu með vongildi, Y . Vongildi margfeldis (X - X )(Y - Y ) er kallað Samdreifni Samdreifni milli X og Y, táknað með Cov(X, Y).Fyrir ósamfelldar hendingar:

An equivalent expression is Jafngild framsetning:

x y

yxYX yxPyxYXEYXCov ),())(()])([(),(

x y

yxyx yxxyPXYEYXCov ),()(),(

Correlation Correlation Fylgni 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:

YX

YXCovYXCorr

),(

),(

Covariance and Statistical Covariance and Statistical IndependenceIndependence

If two random variables are statistically independentstatistically 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.

Portfolio Analysis Portfolio Analysis Greining safnsGreining 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.

bYaXW

Portfolio Analysis Portfolio Analysis Greining Greining safnssafns

The mean value for Wmean value for W is,

The variance for Wvariance for W is,

or using the correlation, skilgreiningu fylgni

YX

W

ba

bYaXEWE

][][

),(222222 YXabCovba YXW

YXYXW YXabCorrba ),(222222

Key WordsKey Words Bernoulli Random

Variable, Mean and Variance

Binomial Distribution Conditional Probability

Function Continuous Random

Variable Correlation Covariance Cumulative Probability

Function

Differences of Random Variables

Discrete Random Variable Expected Value Expected Value: Functions

of Random Variables Expected Value: Function of

Jointly Distributed Random Variable

Hypergeometric Distribution Independence of Jointly

Distributed Random Variables

Key WordsKey Words(continued)(continued)

Joint Probability Function Marginal Probability

Function Mean of Binomial

Distribution Mean: Functions of

Random Variables Poisson Approximation

to the Binomial Distribution

Poisson Distribution Portfolio Analysis

Portfolio, Market Value Probability Distribution

Function Properties: Cumulative

Probability Functions Properties: Joint

Probability Functions Properties: Probability

Distribution Functions Random Variable

Key WordsKey Words(continued)(continued)

Relationships: Probability Function and Cumulative Probability Function

Standard Deviation: Discrete Random Variable

Sums of Random Variables

Variance: Binomial Distribution

Variance: Discrete Random Variable

Variance: Discrete Random Variable (Alternative Formula)

Variance: Functions of Random Variables