Chapter 6Chapter 6

Continuous Random Continuous Random Variables and Variables and

Probability Probability DistributionsDistributions

Samfelldar hendingar og Samfelldar hendingar og líkindadreifingarlíkindadreifingar

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Continuous Random Continuous Random VariablesVariables

Samfelldar hendingar Samfelldar hendingar

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

Cumulative Distribution Cumulative Distribution Function Function DreififallDreififall

The cumulative distribution functioncumulative distribution function, F(x), for a continuous random variable X expresses the probability that X does not exceed the value of x, as a function of x Dreififall, F(x), fyrir samfellda hendingu X gefur til kynna líkurnar á því að X sé lægri en eða jöfn x, sem fall af x

)()( xXPxF

Cumulative Distribution Cumulative Distribution Function Function Dreififall Dreififall

(Figure 6.1)(Figure 6.1)

0 1

1

F(x)

Cumulative Distribution Function for a Random variable Over 0 to 1

Cumulative Distribution Cumulative Distribution Function Function DreififallDreififall

Let X be a continuous random variable with a cumulative distribution function F(x), and let a and b be two possible values of X, with a < b. The probability that X lies probability that X lies between a and bbetween a and b is Látum X vera samfellda hendingu með dreififalli F(x), og látum a og b vera tvö möguleg gildi X þar sem a < b. Líkurnar á því að X liggi milli a og b eru

)()()( aFbFbXaP

Probability Density Function Probability Density Function ÞéttifallÞéttifall

Let X be a continuous random variable, and let x be any number lying in the range of values this random variable can take. The probability density probability density functionfunction, f(x), of the random variable is a function with the following properties:

1. f(x) > 0 for all values of x2. The area under the probability density function f(x)

over all values of the random variable X is equal to 1.0

Látum X vera samfellda hendingu og látum x vera tölu á því bili sem hendingin getur tekið. Þéttifall f(x) hendingarinnar X er fall með eftirfarandi eiginleikum.

1. f(x) > 0 fyrir öll gildi x2. Svæðið undir þéttifallinu f(x) fyrir öll möguleg gildi

hendingarinnnar X er jafnt 1.0

Probability Density FunctionProbability Density Function ÞéttifallÞéttifall

3. Suppose this density function is graphed. Let a and b be two possible values of the random variable X, with a<b. Then the probability that X lies between a and b is the area under the density function between these points.

4. The cumulative density function F(x0) is the area under the probability density function f(x) up to x0

3. Hugsum okkur að ferill þéttifallsins sé teiknaður. Látum a og b vera tvö möguleg gildi á X þar sem a<b. Þá eru líkurnar á því að X liggi á milli a og b svæðið undir þéttifallinu milli þessara tveggja gilda.

4. Dreififallið F(x0) svarar til svæðisins undir þéttifallinu f(x) til og með x0

where xm is the minimum value of the random variable x.þar sem xm lágmarksgildi hendingarinnar x.

SJA BLS 215

0

)()( 0

x

xm

dxxfxF

Shaded Area is the Probability Shaded Area is the Probability That X is Between a and bThat X is Between a and b

(Figure 6.3)(Figure 6.3)

x ba

Probability Density Function for Probability Density Function for a Uniform a Uniform (einsleita)(einsleita) 0 to 1 0 to 1

Random VariableRandom Variable(Figure 6.4)(Figure 6.4)

0 1

1

x

f(x)

Areas Under Continuous Probability Areas Under Continuous Probability Density Functions Density Functions Svæði undir Svæði undir

þéttiföllumþéttiföllum Let X be a continuous random variable with the

probability density function f(x) and cumulative distribution F(x). Then the following properties hold:

1. The total area under the curve f(x) = 1.2. The area under the curve f(x) to the left of x0 is

F(x0), where x0 is any value that the random variable can take.Látum X vera samfellda hendingu með þéttifalli f(x) og dreififalli F(x) þá gilda eftirfarandi eiginleikar fallanna.

Svæðið undir þéttifallinu f(x) vinstra megin við x0 er F(x0), þar sem x0 er sérhvert gildi sem hending getur tekið

1. Heildarsvæði undir þéttifallinu f(x) er 12. The area under the curve f(x) to the left of x0 is

F(x0), where x0 is any value that the random variable can take.

Properties of the Probability Properties of the Probability Density FunctionDensity Function

(Figure 6.6 (a))(Figure 6.6 (a))

0 1 xx0

f(x)

0

1

Comments

Total area under the uniform probability density function is 1.

Aths.Heildarsvæði undir einsleita þéttifallinu er 1.

Properties of the Probability Properties of the Probability Density FunctionDensity Function

(Figure 6.6 (b))(Figure 6.6 (b))

0 1 xx0

f(x)

0

1

Comments

Area under the uniform probability density function to the left of x0 is F(x0), which is equal to x0 for this uniform distribution because f(x)=1.

Rationale for Expectations of Rationale for Expectations of Continuous Random Variables Continuous Random Variables Rök Rök

fyrir vongildi samfelldrar hendingarfyrir vongildi samfelldrar hendingarSuppose that a random experiment leads to an outcome that can be represented by a continuous random variable. If N independent replications of this experiment are carried out, then the expected valueexpected value of the random variable is the average of the values taken, as the number of replications becomes infinitely large. The expected value of a random variable is denoted by E(X).E(X).Hugsum okkur að slembin tilraun geti leitt til niðurstöður sem hægt er setja fram sem samfellda hendingu. Ef N óháðar endurtekningar af þessari tilraun eru framkvæmdar þá er vongildi expected expected valuevalue hendingarinnar meðaltal gildanna sem eru niðurstöður tilraunanna þegar endurtekningarnar verða óendanlega margar. Vongildi samfelldrar hendingar er táknað með E(X).E(X).

Rationale for Expectations of Continuous Rationale for Expectations of Continuous Random Variables Random Variables Rökin fyrir vongildi falls af Rökin fyrir vongildi falls af

samfelldri hendingusamfelldri hendingu (continued) (continued)Similarly, if g(x) is any function of the random variable, X, then the expected value of this function is the average value taken by the function over repeated independent trials, as the number of trials becomes infinitely large. This expectation is denoted E[g(X)]. By using calculus we can define expected values for continuous random variables similarly to that used for discrete random variables. Á sama hátt ef g(x) er fall af hendingu X þá er vongildi þessa falls meðalgildi fallsins sem fæst með því að framkvæma óendanlegan fjölda óháðra tilrauna reikna fallgildi í sérhvert skipti og reikna svo meðaltal fallgildanna sem komu út.. Þetta vongildi er táknað með E[g(X)]. Með því að nota örsmæðarreikning fyrir samfelldar hendingar fyrir öll gildi á X á sama hátt og við notuðum útreikninga fyrir ósamfelldar hendingar:

x

dxxfxgxgE )()()]([

Mean, Variance, and Standard Mean, Variance, and Standard DeviationDeviation

Let X be a continuous random variable. There are two important expected values that are used routinely to define continuous probability distributions.

i. The mean of Xmean of X, denoted by X, is defined as the expected value of X.Látum X vera samfellda hendingu. Það eru tvö mikilvæg vongildi til sem eru

venjubundið notuð til að skilgreina líkindadreifingar samfelldra hendinga.i. Vongildið af X, X, er skilgreint sem

ii. The variance of Xvariance of X, denoted by X2, is defined as the expectation of the

squared deviation, (X - X)2, of a random variable from its meanii. Dreifni X, táknað með X

2, er skilgreint sem vongildi af fráviki hendingar frá meðaltali í öðru veldi, (X - X)2, o

Or an alternative expression can be derived

iii. The standard deviation of Xstandard deviation of X, X, is the square root of the variance.iii. Staðalfrávik hendingar X, X, er kvaðratrótin af dreifni hendingarinnar X.

)(XEX

])[( 22XX XE

222 )( XX XE

Linear Functions of Variables Linear Functions of Variables Línuleg föll af Línuleg föll af hendingumhendingum

Let X be a continuous random variable with mean X and variance X

2, and let a and b any constant fixed numbers. Define the random variable W as Látum X vera samfellda hendingu með vongildi X og dreifni X

2, látum ennfremur a og b vera fastar stærðir. Skilgreinum nú hendingu W sem

Then the mean and variance of W are

and

and the standard deviation of W is

bXaW

XW babXaE )(

222 )( XW bbXaVar

XW b

Linear Functions of VariableLinear Functions of Variable(continued)(continued)

An important special case of the previous results is the standardized random variableMjög mikilvæg niðurstaða fyrrgreindrar glæru er staðlaða hendingin

which has a mean 0 and variance 1.Sem hefur vongildi 0 og dreifni 1.

X

XXZ

Reasons for Using the Reasons for Using the Normal DistributionNormal Distribution

1. The normal distribution closely approximates the probability distributions of a wide range of random variables.

2. Distributions of sample means approach a normal distribution given a “large” sample size.

1. Normal dreifingin er mjög góð nálgun við líkindadreifingar margra breyta.

2. Dreifingar úrtaksmeðaltals nálgast normal dreifinguna að gefnu stóru úrtaki.

Reasons for Using the Normal Reasons for Using the Normal DistributionDistribution

1. The normal distribution closely approximates the probability distributions of a wide range of random variables. Normal dreifingin er mjög góð nálgun við líkindadreifingar margra breyta.

2. Distributions of sample means approach a normal distribution given a “large” sample size. Dreifingar úrtaksmeðaltals nálgast normal dreifinguna að gefnu stóru úrtaki.

3. Computations of probabilities are direct and elegant. Útreikningar á líkindum eru auðveldir og fljótfengnir

4. The normal probability distribution has led to good business decisions for a number of applications. Auðvelt er að nota normal dreifinguna fyrir fjölda viðskiptaákvarðana.

Probability Density Function Probability Density Function for a Normal Distributionfor a Normal Distribution

(Figure 6.8)(Figure 6.8)

x0.0

0.1

0.2

0.3

0.4

Probability Density Function of the Probability Density Function of the Normal Distribution Normal Distribution Þéttifall Þéttifall

normaldreifingarnormaldreifingar

The probability density function for a probability density function for a normally distributed random variable Xnormally distributed random variable X is

Where and 2 are any number such that - < < and - < 2 < and where e and are physical constants, e = 2.71828. . . and = 3.14159. . .

xexf x -for 2

1)(

22 2/)(

2

Properties of the Normal Distribution (188) Properties of the Normal Distribution (188) Eiginleikar normaldreifingarEiginleikar normaldreifingar

Suppose that the random variable X follows a normal distribution with parameters stikum /metlum and 2. Then the following properties hold:

i. The mean of the random variable is ,ii. The variance of the random variable is 2, Dreifni

hendingarinnar er 2,

iii. The shape of the probability density function is a symmetric bell-shaped curve centered on the mean as shown in Figure 6.8. Lögun þéttifallsins er samfelldur bjöllulaga ferill með miðju í eins og sýnt er í mynd 6.8.

iv. By knowing the mean and variance we can define the normal distribution by using the notation Ef við þekkjum vongildi og dreifni normaldreifðar hendingar getum við skilgreint normaldreifinguna sem:

)(XE

22 ])[( XXE

),(~ 2NX

Effects of Effects of on the Probability on the Probability Density Function of a Normal Density Function of a Normal

Random VariableRandom Variable(Figure 6.9 (a))(Figure 6.9 (a))

x

0.0

0.1

0.2

0.3

0.4

1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5

Mean = 5 Mean = 6

Effects of Effects of 22 on the Probability on the Probability Density Function of a Normal Density Function of a Normal

Random VariableRandom Variable(Figure 6.9 (b))(Figure 6.9 (b))

x

0.0

0.1

0.2

0.3

0.4

1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5

Variance = 0.0625

Variance = 1

Cumulative Distribution Function of the Normal Cumulative Distribution Function of the Normal DistributionDistribution Dreififall normaldreifingarDreififall normaldreifingar

Suppose that X is a normal random variable with mean and variance 2 ; that is X~N(, 2). Then the cumulative distribution functioncumulative distribution function is Hugsum okkur að X sé normal dreifð hending með vongildi og dreifni 2 ; þ.e. X~N(, 2). Þá er dreififallið

This is the area under the normal probability density function to the left of x0, as illustrated in Figure 6.10. As for any proper density function, the total area under the curve is 1; that is F() = 1. Þetta er svæðið under ferli þéttifalls normaldreifingar vinstra megin við x0, eins og sýnt er á mynd 6.10. Eins og fyrir öll önnur þéttiföll er heildarsvæðið undir ferli þess 1; þ.e. F() = 1.

)()( 00 xXPxF

Shaded Area is the Probability that X does not Shaded Area is the Probability that X does not Exceed xExceed x00 for a Normal Random Variable for a Normal Random Variable

Skyggða svæðið svarar til líkanna á því að að X sé Skyggða svæðið svarar til líkanna á því að að X sé ekki hærri en x0 fyrir normaldreifða hendinguekki hærri en x0 fyrir normaldreifða hendingu

(Figure 6.10)(Figure 6.10)

xx0

f(x)

Range Probabilities for Normal Random Range Probabilities for Normal Random Variables Variables Líkur á bili fyrir normaldreifða Líkur á bili fyrir normaldreifða

hendinguhendingu

Let X be a normal random variable with cumulative distribution function F(x), and let a and b be two possible values of X, with a < b. ThenLátum X vera normaldreifða hendingu með dreififalli F(x), látum ennfremur a og b vera tvö möguleg gildi á X, með a < b. Þá gildir

The probability is the area under the corresponding probability density function between a and b.Líkurnar eru svæðið undir ferli þéttifallsins á bilinu á milli a og b.

)()()( aFbFbXaP

Range Probabilities for Normal Random Range Probabilities for Normal Random VariablesVariables

Líkur á bili fyrir normaldreifinguLíkur á bili fyrir normaldreifingu(Figure 6.12)(Figure 6.12)

xb

f(x)

a

The Standard Normal DistributionThe Standard Normal Distribution Hin staðlaða normaldreifingHin staðlaða normaldreifing

Let Z be a normal random variable with mean 0 and variance 1; that isLátum Z vera normaldreifingu með vongildi 0 og dreifni 1 þ.e.;

We say that Z follows the standard normal distribution. Denote the cumulative distribution function as F(z), and a and b as two numbers with a < b, thenVið segjum að Z fylgi staðlaðri normaldreifingu. Táknum dreififall þess sem F(z) og látum a og b vera tvær tölur með a < b, þá gildir

)1,0(~ NZ

)()()( aFbFbZaP

Standard Normal Distribution with Standard Normal Distribution with Probability for Probability for

Stöðluð normaldreifing og líkur meðStöðluð normaldreifing og líkur með z = z = 1.251.25

(Figure 6.13)(Figure 6.13)

z 1.25

0.8944

Standard Normal DistributionStandard Normal Distribution

Standard Normal DistributionStandard Normal Distribution

Finding Range Probabilities for Normally Finding Range Probabilities for Normally Distributed Random Variables Distributed Random Variables Að finna líkur á Að finna líkur á

bili fyrir normaldreifðar hendingarbili fyrir normaldreifðar hendingarLet X be a normally distributed random variable with mean and variance 2. Then the random variable Z = (X - )/ has a standard normal distribution: Z ~ N(0, 1)It follows that if a and b are any numbers with a < b, thenLátum X vera normaldreifða hendingu með vongildi og dreifni 2. Þá er normaldreifða hendingin Z = (X - )/ með staðlaða normaldreifingu : Z ~ N(0, 1)Samkvæmt því gildir að ef a og b eru einhverjar tölur þar sem a < b, þá gildir

where Z is the standard normal random variable and F(z) denotes its cumulative distribution function.Þar sem Z er stöðluð normaldreifð hending og F(z) er dreififall hennar.

aF

bF

bZ

aPbXaP )(

Computing Normal Probabilities Computing Normal Probabilities Útreikningar með normaldreifinguÚtreikningar með normaldreifingu

(Example 6.6)(Example 6.6)

A very large group of students obtains test scores that are normally distributed with mean 60 and standard deviation 15. What proportion of the students obtained scores between 85 and 95? Stór hópur nemenda fær einkunnir sem eru normaldreifðar með vongildi 60 og staðalfráviki 15. Hversu stórt hlutfall nemenda verður með einkunn á milli 85 og 95?

0376.09525.09901.0

)67.1()33.2(

)33.267.1(

15

6095

15

6085)9585(

FF

ZP

ZPXP

That is, 3.76% of the students obtained scores in the range 85 to 95.

Approximating Binomial Probabilities Using the Approximating Binomial Probabilities Using the Normal Distribution Normal Distribution Nálgun tvíliðunardreifingar með Nálgun tvíliðunardreifingar með

normaldreifingunormaldreifingu

Let X be the number of successes from n independent Bernoulli trials, each with probability of success . The number of successes, X, is a Binomial random variable and if n(1 - ) > 9 a good approximation isLátum X vera fjölda heppnaðara atburða úr n óháðum Bernoulli tilraunum þar sem sérhver hefur líkurnar á að heppnast. Fjöldi heppnaðra tilrauna X er tvíliðunardreifð hending og ef n(1 - ) > 9 þá er góð nálgun fengin með

Or if 5 < n(1 - ) < 9 we can use the continuity correction factor to obtain

where Z is a standard normal variable. Þar sem Z er stöðluð normaldreifð hending.

)1()1()(

n

nbZ

n

naPbXaP

)1(

5.0

)1(

5.0)(

n

nbZ

n

naPbXaP

The Exponential DistributionThe Exponential DistributionThe exponential random variable T (t>0) has a probability density function Exponential Hendingin T (t>0) hefur þéttifall

Where is the mean number of occurrences per unit time, t is the number of time units until the next occurrence, and e = 2.71828. . . Then T is said to follow an exponential an exponential probability distributionprobability distribution.

The cumulative distribution function is Þar sem er meðalfjöldi atburða yfir einingu tíma, t er fjöldi tímaeininga þar til næsti atburður á sér stað, og e = 2.71828. . . Þá er T sagt fylgja exponential líkindadreifingu.

Dreififallið er

The distribution has mean 1/ and variance 1/2

Hendingin hefur vongildi 1/ og dreifni 1/2

0 for t )( tetf

0 for t 1)( tetF

Probability Density Function for Probability Density Function for an Exponential Distribution with an Exponential Distribution with

= 0.2 = 0.2(Figure 6.27)(Figure 6.27)

x

f(x)

Lambda = 0.2

0 10 20

0.0

0.1

0.2

Joint Cumulative Distribution Joint Cumulative Distribution FunctionsFunctions

Let X1, X2, . . .Xk be continuous random variablesi. Their joint cumulative distribution functioncumulative distribution function, F(x1,

x2, . . .xk) defines the probability that simultaneously X1 is less than x1, X2 is less than x2, and so on; that is Látum X1, X2, . . .Xk vera samfelldar hendingar Sameiginlegt dreififall þeirra, F(x1, x2, . . .xk) skilgreinir líkurnar á því að á sama tíma sé X1 minna en x1, X2 minna en x2, o.s.frv. Þ.e.:

ii. The cumulative distribution functions F(x1), F(x2), . . .,F(xk) of the individual random variables are called their marginal marginal distribution functionsdistribution functions. For any i, F(xi) is the probability that the random variable Xi does not exceed the specific value xi. Dreififöllin F(x1), F(x2), . . .,F(xk) einstakra hendinga eru kölluð jaðardreififöll marginal distribution functionsmarginal distribution functions. Fyrir sérhvert i, F(xi) er líkurnar á því að hendingin Xi sé ekki stærri en xi.

iii. The random variables are independentindependent if and only if Hendingarnar eru óháðar eff

)(),,,( 221121 kkk xXxXxXPxxxF

)()()(),,,( 2121 kk xFxFxFxxxF

Marginal Distribution for Bivariate Marginal Distribution for Bivariate DistributionDistribution

Látum X og Y vera sundurleitar hendingar (discreate random variables)

Bivariate distribution með n pörum (pairs) gilda (values) má setja fram sem:

(x1,y1), (x2,y2),(x3,y3),...,(xn,yn)Jaðardreifingu X má skrifa sem

Jaðardreifingu Y má skrifa sem

kkjj yYxXPxXP )&()(

j

kjk yYxXPyYP )&()(

CovarianceCovarianceLet X and Y be a pair of continuous random variables, with respective means x and y. The expected value of (x - x)(Y - y) is called the covariancecovariance between X and Y. That is Látum X og Y vera pör samfelldra hendinga með vongildi x og y. Vongildið af (x - x)(Y - y) er kallað samdreifni milli X og Y. Það er:

An alternative but equivalent expression can be derived as Annars konar framsetning en jafnframt jafngild má leiða út sem

If the random variables X and Y are independent, then the covariance between them is 0. However, the converse is not true (i.e. if the convariance is zero, they are not necessarily independent).

)])([(),( yx YXEYXCov

yxXYEYXCov )(),(

Correlation Correlation FylgniFylgni Let X and Y be jointly distributed random variables. The correlationcorrelation between X and Y is Látum X og Y vera sameiginlega dreifðar hendingar. Fylgni milli X og Y er skilgreind sem:

Rho hleypur milli -1 og 1. Ef fylgni milli hendinganna (random variables) X og Y er 1 eða -1, þá felur það í sér að setja má fram línulega sambandið Y=a + bX, þar sem a og b eru fastar. Ef b>0 => rho=1 Ef b <0 => rho=-1. Ef X og y eru óháðar (completely unrelated / independent) þá er rho=0. Ef rho=0 =>X og Y eru sagðar vera uncorrelated. En rho túlkar aðeins línulegt samband og rho=0 felur það ekki í sér að X og Y séu alltaf óháðað (independent).

YX

YXCovYXCorr

),(

),(

Sums of Random VariablesSums of Random Variables

Let X1, X2, . . .Xk be k random variables with means 1, 2,. . . k and variances 1

2, 22,. . ., k

2. The following properties hold: Látum X1, X2, . . .Xk vera k hendingar með vongildi 1, 2,. . . k og dreifni 1

2, 22,. . ., k

2. Eftirfarandi eiginleikar gilda þá:

i. The mean of their sum is the sum of their means; that is

ii. If the covariance between every pair of these random variables is 0, then the variance of their sum is the sum of their variances; that is Ef samdreifni milli sérhverrar hendingar og hinna hendinganna er 0 þá gildir:

However, if the covariances between pairs of random variables are not 0, the variance of their sum is

kkXXXE 2121 )(

222

2121 )( kkXXXVar

),(2)(1

1 1

222

2121 j

K

i

K

ijikk XXCovXXXVar

Differences Between a Pair of Random Differences Between a Pair of Random Variables Variables Mismunur milli para hendingaMismunur milli para hendinga

Let X and Y be a pair of random variables with means X and Y and variances X

2 and Y2. Then following properties hold:

Látum X og Y vera pör af hendingum með vongildi X og Y og dreifni X2 og Y2. Þá gilda eftirfarandi eiginleikar:

i. The mean of their difference is the difference of their means; that is Vongildi mismunar er mismunur vongilda

ii. If the covariance between X and Y is 0, then the variance of their difference is Ef samdreifni milli X og Y er 0 þá er dreifni mismunar summa dreifninnar

iii. If the covariance between X and Y is not 0, then the variance of their difference is Ef samdreifni milli X og Y er ekki 0 þá er dreifni mismunar

YXYXE )(

22)( YXYXVar

),(2)( 22 YXCovYXVar YX

Linear Combinations of Random Variables Linear Combinations of Random Variables Línuleg samantekt hendingaLínuleg samantekt hendinga

The linear combination of two random variables, X and Y, is Línuleg samantekt hendinga, X og Y, er

Where a and b are constant numbers. The mean for W is, Þar sem a og b eru fastar. Vongildi W er,

The variance for W is, Dreifni W er,

Or using the correlation, Eða með því að nota fylgni

If both X and Y are joint normally distributed random variables then the resulting random variable, W, is also normally distributed with mean and variance derived above. Ef bæði X og Y eru sameiginlega normaldreifðar hendingar þá verður W normaldreifð með vongildi og dreifni eins og greinir að ofan.

bYaXW

YXW babYaXEWE ][][

),(222222 YXabCovba YXW

YXYXW YXabCorrba ),(222222

Key WordsKey Words Approximating Binomial

Probabilities Using the Normal Distribution

Area Under Continuous Probability Density Functions

Correlation Covariance Cumulative Distribution

Function Cumulative Distribution

Function of the Normal Distribution

Differences Between Pairs of Random Variable

Expectations of Continuous Random Variables

Exponential Distribution Finding Range Probabilities

for Normal Random Variables Joint Cumulative Distribution

Function Linear Combinations of

Random variables Linear Functions of Random

Variables

Key WordsKey Words(continued)(continued)

Mean of a Continuous Random Variable

Probability Density Function

Probability Density Function of the Normal Distribution

Properties of a Normal Distribution

Range Probabilities for Normal Random Variables

Range Probabilities Using a Cumulative Distribution Function

Standard Deviation: Continuous Random Variable

Standard Normal Distribution

Sums of Random Variables

Uniform Distribution Variance