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Chapter 6

Chapter 6. Continuous Random Variables and Probability Distributions Samfelldar hendingar og líkindadreifingar. Continuous Random Variables Samfelldar hendingar.

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Chapter 6

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  1. Chapter 6 Continuous Random Variables and Probability Distributions Samfelldar hendingar og líkindadreifingar

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

  3. Cumulative Distribution Function Dreififall The cumulative 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

  4. Cumulative Distribution Function Dreififall (Figure 6.1) F(x) 1 0 1 Cumulative Distribution Function for a Random variable Over 0 to 1

  5. Cumulative Distribution Function Dreififall 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 between 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

  6. Probability Density Function Þéttifall • 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 function, f(x), of the random variable is a function with the following properties: • f(x) > 0 for all values of x • 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. • f(x) > 0 fyrir öll gildi x • Svæðið undir þéttifallinu f(x)fyrir öll möguleg gildi hendingarinnnar X er jafnt 1.0

  7. Probability Density FunctionÞéttifall 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. • The cumulative density function F(x0) is the area under the probability density function f(x) up to x0 • 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. • 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

  8. Shaded Area is the Probability That X is Between a and b(Figure 6.3) 0 a b x

  9. Probability Density Function for a Uniform (einsleita) 0 to 1 Random Variable(Figure 6.4) f(x) 1 0 1 x

  10. Areas Under Continuous Probability Density Functions Svæði undir þéttiföllum • Let X be a continuous random variable with the probability density function f(x) and cumulative distribution F(x). Then the following properties hold: • The total area under the curve f(x) = 1. • 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 • Heildarsvæði undir þéttifallinu f(x) er 1 • 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.

  11. Properties of the Probability Density Function(Figure 6.6 (a)) Comments Total area under the uniform probability density function is 1. Aths. Heildarsvæði undir einsleita þéttifallinu er 1. f(x) 1 0 0 x0 1 x

  12. Properties of the Probability Density Function(Figure 6.6 (b)) Comments Area under the uniform probability density function to the left of x0 is F(x0), which is equal to x0for this uniform distribution because f(x)=1. f(x) 1 0 0 x0 1 x

  13. Rationale for Expectations of Continuous Random Variables Rök fyrir vongildi samfelldrar hendingar Suppose 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 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). 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 value hendingarinnar meðaltal gildanna sem eru niðurstöður tilraunanna þegar endurtekningarnar verða óendanlega margar. Vongildi samfelldrar hendingar er táknað með E(X).

  14. Rationale for Expectations of Continuous Random Variables Rökin fyrir vongildi falls af samfelldri hendingu (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:

  15. Mean, Variance, and Standard Deviation Let X be a continuous random variable. There are two important expected values that are used routinely to define continuous probability distributions. • The mean 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. • Vongildið af X, X, er skilgreint sem • The variance of X, denoted by X2, is defined as the expectation of the squared deviation, (X - X)2, of a random variable from its mean ii. Dreifni X, táknað með X2, er skilgreint sem vongildi af fráviki hendingar frá meðaltali í öðru veldi, (X - X)2, o Or an alternative expression can be derived • The standard deviation of X, X, is the square root of the variance. iii. Staðalfrávik hendingar X, X, er kvaðratrótin af dreifni hendingarinnar X.

  16. Linear Functions of Variables Línuleg föll af hendingum Let X be a continuous random variable with mean X and variance X2, 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 X2, 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

  17. Linear Functions of Variable(continued) An important special case of the previous results is the standardized random variable Mjög mikilvæg niðurstaða fyrrgreindrar glæru er staðlaða hendingin which has a mean 0 and variance 1. Sem hefur vongildi 0 og dreifni 1.

  18. Reasons for Using the Normal Distribution • The normal distribution closely approximates the probability distributions of a wide range of random variables. • 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.

  19. Reasons for Using the Normal Distribution • 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. • Distributions of sample means approach a normal distribution given a “large” sample size. Dreifingar úrtaksmeðaltals nálgast normal dreifinguna að gefnu stóru úrtaki. • Computations of probabilities are direct and elegant. Útreikningar á líkindum eru auðveldir og fljótfengnir • The normal probability distribution has led to good business decisions for a number of applications. Auðvelt er að nota normal dreifinguna fyrir fjölda viðskiptaákvarðana.

  20. Probability Density Function for a Normal Distribution(Figure 6.8) 0.4 0.3 0.2 0.1 0.0  x

  21. Probability Density Function of the Normal Distribution Þéttifall normaldreifingar The probability density function for a normally 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. . .

  22. Properties of the Normal Distribution (188) Eiginleikar normaldreifingar Suppose that the random variable X follows a normal distribution with parameters stikum /metlum and 2. Then the following properties hold: • The mean of the random variable is , • The variance of the random variable is 2, Dreifni hendingarinnar er 2, • 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. • 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:

  23. Effects of  on the Probability Density Function of a Normal Random Variable(Figure 6.9 (a)) 0.4 Mean = 6 Mean = 5 0.3 0.2 0.1 0.0 x 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5

  24. Effects of 2 on the Probability Density Function of a Normal Random Variable(Figure 6.9 (b)) 0.4 Variance = 0.0625 0.3 0.2 Variance = 1 0.1 0.0 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 x

  25. Cumulative Distribution Function of the Normal DistributionDreififall normaldreifingar Suppose that X is a normal random variable with mean  and variance 2 ; that is X~N(, 2). Then the cumulative 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.

  26. Shaded Area is the Probability that X does not Exceed x0 for a Normal Random Variable Skyggða svæðið svarar til líkanna á því að að X sé ekki hærri en x0 fyrir normaldreifða hendingu(Figure 6.10) f(x) x0 x

  27. Range Probabilities for Normal Random Variables Líkur á bili fyrir normaldreifða hendingu 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. Then Látum 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.

  28. Range Probabilities for Normal Random VariablesLíkur á bili fyrir normaldreifingu(Figure 6.12) f(x) a b x 

  29. The Standard Normal DistributionHin staðlaða normaldreifing Let Z be a normal random variable with mean 0 and variance 1; that is Látum 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, then Við 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

  30. Standard Normal Distribution with Probability for Stöðluð normaldreifing og líkur meðz = 1.25(Figure 6.13) 0.8944 z 1.25

  31. Standard Normal Distribution

  32. Standard Normal Distribution

  33. Finding Range Probabilities for Normally Distributed Random Variables Að finna líkur á bili fyrir normaldreifðar hendingar Let 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, then Látum 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.

  34. Computing Normal Probabilities Útreikningar með normaldreifingu(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? That is, 3.76% of the students obtained scores in the range 85 to 95.

  35. Approximating Binomial Probabilities Using the Normal Distribution Nálgun tvíliðunardreifingar með normaldreifingu 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 is Látum 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.

  36. The Exponential Distribution The 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 probability 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

  37. Probability Density Function for an Exponential Distribution with  = 0.2(Figure 6.27) f(x) Lambda = 0.2 0.2 0.1 0.0 x 0 10 20

  38. Joint Cumulative Distribution Functions Let X1, X2, . . .Xk be continuous random variables • Their joint cumulative 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 distribution 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 functions. Fyrir sérhvert i, F(xi) er líkurnar á því að hendingin Xi sé ekki stærri en xi. • The random variables are independent if and only if Hendingarnar eru óháðar eff

  39. Marginal Distribution for Bivariate Distribution 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

  40. Covariance Let 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 covariance 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).

  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 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).

  42. Sums of Random Variables Let X1, X2, . . .Xk be k random variables with means 1, 2,. . . k and variances 12, 22,. . ., k2. The following properties hold: Látum X1, X2, . . .Xk vera k hendingar með vongildi 1, 2,. . . k og dreifni 12, 22,. . ., k2. Eftirfarandi eiginleikar gilda þá: • The mean of their sum is the sum of their means; that is • 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

  43. Differences Between a Pair of Random Variables Mismunur milli para hendinga Let X and Y be a pair of random variables with means X and Y and variances X2 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: • The mean of their difference is the difference of their means; that is Vongildi mismunar er mismunur vongilda • 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 • 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

  44. Linear Combinations of Random Variables Línuleg 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.

  45. 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 Words

  46. 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 Key Words(continued)

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