2009 Winton 1 Distributi ( ons 2) (2)

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1 Distributions ib i (2)

2 2 IV. Triangular Distribution ib ti Known values The minimum (a) The mode (b - the most likely value of the pdf) The maimum (c) f() probability density function (area under the curve = ); the area under the triangle () is half the area of its bounding bo which is h(c-a) h 2 c -a b a b c

3 3 pdf for Triangular Distribution The pdf is given by f() 2( - a) (c - a)(b - a) 2( - c) (c - a)(c - b) for a b for b c (slope (slope h ) b - a h ) c - b = otherwise

4 4 Epected Value The epected value is given by E(X) - f()d b a 2( - a) (c - a)(b - a) d c b 2(c - ) (c - a)(c - b) d a b 3 Derivation: with a little work the integrals evaluate to c b - 3ab a c - 3cb 2b E(X) h 6(b - a) 6(c - b) (a b c)(c - a)(b - a)(c - b) a b 3(c - a)(b - a)(c - b) 3 3 c a (c - b) b (a - c) c (b - a) 3(c - a)(b - a)(c - b)

5 5 About mean, mode, and median For a discrete sample, measures of centrality that are typically determined are the mean, the mode, and the median The mean is the average value of the sample and corresponds to E(X) The mode corresponds to the maimum value of the pdf When working with a sample, it is necessary to resort to a histogram (which can be tricky) to estimate the mode of the underlying gpdf The median simply corresponds to that point at which half of the area under the curve is to the left and half is to the right The triangular distribution is typically employed when not much is known about the distribution, but the minimum, mode, and maimum can be estimated

6 Sampling From the Triangular 6 Again this means solving Distribution rsample for rsample given a random probability - f(z)dz Since f(z) is piecewise continuous, its distribution function F(t) is given by F(t) t f(z)dz t a f(z)dz - c t f(z)dz for t for a for b for t a t c t b c Left side integral Right side integral

7 7 Lft Left and drihtsid Right Side Integrals It For left side a rsample b rsample f(z)dz a rsample a 2(z a) (b a)(c a) dz 2 z - 2az (b - a)(c - a) rsample a ( rsample - a) (b - a)(c - a) 2 c For right side b rsample c (c - rsample) - (c - b)(c - a) since c 2 2(c z) 2cz - z (c - rsample) f(z)dz dz (c b)(c a) (c - b)(c - a) (c - b)(c - a) rsample rsample 2 c rsample 2

8 8 b a Sampling Function (b - a) Since f(z)dz (area of left triangle) (c - a) (b - a) if use the left side equation to (c - a) solve for rsample; otherwise use the right side equation rsample a (b - a)(c - a) for (b - a)/(c - a) rsample c - (c - b)(c - a)(- ) for (b - a)/(c - a)

9 9 Graph of Sampling Function: Triangular Distribution rsample Eample: the median corresponds to =.5) For a=, b = 2, c = 4 mean = (a+b+c)/3 = mode = 2 median = c (c b)(c a)(.5) c b a (b-a)/(c-a)

10 V. Gamma Distribution Background: Gamma Function One of a large number of functions related to the eponential function Traces from 8 th century work by Euler in which he was using interpolation methods to define n! for nonintegral values dubbed the gamma function by LeGendre in a series of books published between 8 and 826 Appears naturally in the study of anti-differentiation Also studied in the contet of differential equations when calculating LaPlace transforms The gamma function is given by Γ() ) - e d ( )

11 Gamma Function Definition The gamma function is given by Γ() - e d ( Integrate (by parts) to get () = (-)(-) for > () e - d When is an integer >, () = ( -)! The gamma function is a generalization of the factorial, applying to all >, not just integers )

12 2 Obtaining the Gamma Distribution The gamma distribution is obtained from the gamma function by specifying the pdf f() k - e β for otherwise for fied > and > where the proportionality p constant k is chosen so that f()d.

13 3 Proportionality Constant k so - - β f()d. k e d kβ t e dt k β () and f() is given by - () ) β -t - where = t e -/β is called shape (or order) parameter us called the scale parameter - β When =, f() = e the eponential distribution with mean β In general, E(X) = and 2 = 2 If the mean and standard deviation can be estimated then and can also be determined

14 4 Gamma Function Reflection Formula ( ) - e - d ( ) The general relationship () ( ) = (-)(-) ( ) for > holds It can also be shown that for < < π ( ) (- ) sin(π ) (.5) π For < <, + >, so (+) ) = () ) This gives the reflection formula π (- ) ) for < < Γ( )sin(π )

15 Algorithm for Calculating the Natural 5 Logarithm of the Gamma Function Attributed to Lanczos, C., Journal ls.i.a.m. Numerical lanalysis, ser. B, vol., p. 86 (964) and adapted from Numerical Recipes in C by Press, W.H., and B.P. Flannery, S.A. Teukolsky, W.T. Vetterling (Cambridge University Press, 988) FUNCTION lngamma(z) These values are the IF z < // use the reflection formula for z < (approimate) coefficients for the first 6 terms of an z - z RETURN ln(z) - (lngamma( + z) + ln(sin(z)) ENDIF coeff , , , ,.28583, a FOR i TO6 a a + coeff(i)/(i + z - ) ENDFOR RETURN ln(a) - (z+45)+(z 4.5) + - 5)ln(z.5)ln(z + 4.5) END infinite series involved in an eact formulation for the gamma function credited to Lanczos. They yield an approimation for the variable "a" (determined net) which is within < 2 - of its true value

16 6 Selected Values of () ( ) computed according to the algorithm for ln(()) () (.25) (.5) = π (.75) () =! = (.25) ( ) (.5) =.5(.5) = π / (.75) (2) =! = (2.25).333 (2.5) =5(.5(.5) 5) = 3 π / (2.75) (3) = 2! = 2!! 2! 3! (3.25) (3.5) = 2.5(2.5) = 5 π / (3.75) (4) = 3! = 6 (4.25) (4.5) = 3.5(3.5) = 5 π / (4.75) ( ) (5) = 4! = 24 (5.25) (5.5) = 4.5(4.5) = π / (5.75) (6) =5! = 2

17 7 f().6 Graph of Gamma pdf fied mean = 5 and varying values of and =.5, = =.5, = =5, = =, =.55. 5

18 Corresponding Distribution 8 F() Functions and Sampling Functions rsample The gamma distribution is used to model.2.6 waiting times or time to complete a task It can be shown that if we have eponentially distributed interarrival times with mean /, the time needed to obtain k changes distributes according to a gamma distribution with = k and = / 2

19 9 Gamma Distribution Behavior () () f() k - e β for otherwise for fied > and > If < then - as = then the distribution is the eponential distribution > then - as The eample showed three basic shapes, each of which is described dby the behavior of the derivative f'( '() (slope function) of f() f '() = k[( - ) -2 e -/ +(-/) - e -/ ]

20 2 Gamma Distribution Behavior (2) () - β f() k e for for fied > and > otherwise There are actually 5 cases: < the slope - as since each term is < and each eponent of is < = the slope -/ 2 as in accord with the eponential distribution since k = /, term is and term 2 is -/ < 2 and > the slope + as since the lead term + and dterm 2is = 2 the slope +/ 2 as since k = / = / 2 > 2 the slope as In each case the slope as + The gamma distribution is one which is usually sampled by the accept- reject technique, which means to get k, the value of ()mustbe computed

21 2 VI. Erlang Distribution Erlang was a Danish telephone engineer who did some of the early work in queuing theory For the Gamma distribution, when is an integer i, then the gamma distribution is called an Erlang distribution of order i The Erlang distribution is used to model phenomena having i stages, each with independent, eponentially distributed service times of mean Rather than model these separately, an Erlang distribution of order i can be used to model the total service time e.g., if an eperiment has 3 successive stages each of which takes an average of 5 minutes (eponentially distributed), then the eperiment time can be modeled by taking = 3 and = 5 (to get the overall mean of = 5 minutes)

22 22 VII. Weibull Distribution For the gamma distribution the idea was to choose k so that k is equal to and then choose f() = k - e -/ A major difficulty with this pdf is that it is not integrable in closed form; however, it is close to being so The idea is to render the part under the integral to be in the form e z dz so that the function will be integrable; i.e., for k e /β? - d - we need to determine the? value so that for z = -(/)?, dz= - d If z = -(/) then dz = -(/)(/) - d and then -ββ -ββ - /β /β e /β d k e β β k when k k β e -/β d

23 23 Weibull pdf The choice of k=/ is the basis for the pdf δ f() β β - e - δ - β for δ otherwise the effect of is to displace the distribution along the horizontal ais, so it is often taken to be With a little work, it can be shown that E(X) = + ( + /) and VAR(X) = 2 (( + 2/) - 2 ( + /))

24 24 Sampling Since rsample δ f(z)dz - e - rsample - δ /β - ( rsample - δ)/β - e the sample function can be solved for rsample for given values of,, The Weibull distribution is often used to model time until failure (for eample, light bulbs may have significant early failure and some have significant long term until failure) When =, = /, = then the Weibull distribution is the eponential distribution δ

25 25 Graph f() Weibull pdf for fied = and varying values of =.5, = =, = =5, = =.5, =

System Simulation Part II: Mathematical and Statistical Models Chapter 5: Statistical Models

System Simulation Part II: Mathematical and Statistical Models Chapter 5: Statistical Models Fatih Cavdur fatihcavdur@uludag.edu.tr March 29, 2014 Introduction Introduction The world of the model-builder