Chapter 7. Random-Variate Generation 7.1. Prof. Dr. Mesut Güneş Ch. 7 Random-Variate Generation

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1 Chapter 7 Random-Varate Generaton 7.

2 Contents Inverse-transform Technque Acceptance-Rejecton Technque Specal Propertes 7.

3 Purpose & Overvew Develop understandng of generatng samples from a specfed dstrbuton as nput to a smulaton model. Illustrate some wdely-used technques for generatng random varates: Inverse-transform technque Acceptance-rejecton technque Specal propertes 7.3

4 Preparaton It s assumed that a source of unform [0,] random numbers ests. Lnear Congruental Method (LCM) Random numbers R, R, R, wth PDF CDF f R F R ( ) 0 0 ( ) 0 otherwse < 0 0 > f() 0 F() 0 7.4

5 Inverse-transform Technque 7.5

6 Inverse-transform Technque The concept: For CDF functon: r F() Generate r from unform (0,), a.k.a U(0,) Fnd, F - (r) F() F() r F() r F() r r r 7.6

7 Inverse-transform Technque The nverse-transform technque can be used n prncple for any dstrbuton. Most useful when the CDF F() has an nverse F - () whch s easy to compute. Requred steps. Compute the CDF of the desred random varable. Set F() R on the range of 3. Solve the equaton F() R for n terms of R 4. Generate unform random numbers R, R, R 3,... and compute the desred random varate by F - (R ) 7.7

8 Inverse-transform Technque: Eample Eponental Dstrbuton PDF CDF f ( ) F( ) λe λ e Smplfcaton ln(r) λ λ Snce R and (-R) are unformly dstrbuted on [0,] To generate,, 3 e e λ λ λ R R ln( R) ln( R) λ ln( R) λ F ( R) 7.8

9 Inverse-transform Technque: Eample 7.9

10 Inverse-transform Technque: Eample Inverse-transform technque for ep(λ ) 7.0

11 Inverse-transform Technque: Eample Eample: Generate 00 or 500 varates wth dstrbuton ep(λ ) Generate 00 or 500 R s wth U(0,), the hstogram of s becomes: 0,7 0,6 0,6 0,5 0,5 0,4 0,4 0,3 0,3 0, 0, 0, 0, 0 0,5,5,5 3 3,5 4 4,5 5 5,5 6 6,5 7 Emprcal Hstogram 0 0,57,5,7,30,87 3,45 4,0 4,60 5,7 5,75 Rel Prob. Theor. PDF 7.

12 Inverse-transform Technque Check: Does the random varable have the desred dstrbuton? P ( 0) P( R F( 0)) F( 0) 7.

13 Inverse-transform Technque: Other Dstrbutons Eamples of other dstrbutons for whch nverse CDF works are: Unform dstrbuton Webull dstrbuton Trangular dstrbuton 7.3

14 7.4 Inverse-transform Technque: Unform Dstrbuton Random varable unformly dstrbuted over [a, b] ) ( ) ( ) ( a b R a a b R a R a b a R F +

15 Inverse-transform Technque: Webull Dstrbuton The Webull Dstrbuton s descrbed by PDF CDF f ( ) F( ) β β α β e e ( ) β α ( ) β α The varate s e F( ) e ( ) ( ) ( ) α β α β α α β β β β R R R ln( ln( α β R) R) ln( R) β α α β β ln( ln( R) R) 7.5

16 7.6 Inverse-transform Technque: Trangular Dstrbuton The CDF of a Trangular Dstrbuton wth endponts (0, ) s gven by s generated by < ) ( 0 R R R R ) ( 0 ) ( R > < < ) ( ) ( F

17 Inverse-transform Technque: Emprcal Contnuous Dstrbutons When theoretcal dstrbutons are not applcable To collect emprcal data: Resample the observed data Interpolate between observed data ponts to fll n the gaps 7.7

18 Inverse-transform Technque: Emprcal Contnuous Dstrbutons For a small sample set (sze n): Arrange the data from smallest to largest Set (0) 0 () () (n) Assgn the probablty /n to each nterval The slope of each lne segment s defned as a The nverse CDF s gven by ) ( ) ( ) n n Fˆ ( R) ( ) ( ( ) ( ) n + a R ( ) n (-) (),,,n when ( ) < n R n 7.8

19 Inverse-transform Technque: Emprcal Contnuous Dstrbutons Interval PDF CDF Slope a 0.0 < < < < < R 0.7 (4) + a 4 ( R (4 ) / ( ).66 n) 7.9

20 Inverse-transform Technque: Emprcal Contnuous Dstrbutons What happens for large samples of data Several hundreds or tens of thousand Frst summarze the data nto a frequency dstrbuton wth smaller number of ntervals Afterwards, ft contnuous emprcal CDF to the frequency dstrbuton Slght modfcatons Slope a ( ) c c ( ) The nverse CDF s gven by Fˆ ( R) + a R c c cumulatve probablty of the frst ntervals ( ) when c < R c ( ) 7.0

21 Inverse-transform Technque: Emprcal Contnuous Dstrbutons Eample: Suppose the data collected for 00 broken-wdget repar tmes are: Interval (Hours) Frequency Relatve Frequency Cumulatve Frequency, c Slope, a Consder R 0.83: c < R < c 4.00 (4-) + a 4 (R c (4-) ) ( ).75 7.

22 Inverse-transform Technque: Emprcal Contnuous Dstrbutons Problems wth emprcal dstrbutons The data n the prevous eample s restrcted n the range The underlyng dstrbuton mght have a wder range Thus, try to fnd a theoretcal dstrbuton Hnts for buldng emprcal dstrbutons based on frequency tables It s recommended to use relatvely short ntervals Number of bns ncrease Ths wll result n a more accurate estmate 7.

23 Inverse-transform Technque: Contnuous Dstrbutons A number of contnuous dstrbutons do not have a closed form epresson for ther CDF, e.g. Normal Gamma F( ) ( ( ) tµ )dt ep σ π σ Beta The presented method does not work for these dstrbutons Soluton Appromate the CDF or numercally ntegrate the CDF Problem Computatonally slow 7.3

24 Inverse-transform Technque: Dscrete Dstrbuton All dscrete dstrbutons can be generated va nversetransform technque Method: numercally, table-lookup procedure, algebracally, or a formula Eamples of applcaton: Emprcal Dscrete unform Geometrc 7.4

25 Inverse-transform Technque: Dscrete Dstrbuton Eample: Suppose the number of shpments,, on the loadng dock of a company s ether 0,, or Data - Probablty dstrbuton: P() F() The nverse-transform technque as table-lookup procedure F ( ) r < R r F( ) Set 7.5

26 Inverse-transform Technque: Dscrete Dstrbuton Method - Gven R, the generaton scheme becomes: 0,,, 0.5 < 0.8 < R 0.5 R 0.8 R Table for generatng the dscrete varate Input r Output Consder R 0.73: F( - ) < R F( ) F( 0 ) < 0.73 F( ) Hence, 7.6

27 Acceptance-Rejecton Technque 7.7

28 Acceptance-Rejecton Technque Useful partcularly when nverse CDF does not est n closed form Thnnng Illustraton: To generate random varates, ~ U(/4,) Procedure: Step. Generate R ~ U(0,) Step. If R ¼, accept R. Step 3. If R < ¼, reject R, return to Step no Generate R Condton yes Output R R does not have the desred dstrbuton, but R condtoned (R ) on the event {R ¼} does. Effcency: Depends heavly on the ablty to mnmze the number of rejectons. 7.8

29 Acceptance-Rejecton Technque: Posson Dstrbuton Probablty mass functon of a Posson Dstrbuton n α α P( N n) e n! Eactly n arrvals durng one tme unt A + A + + An < A + A + + An + An + Snce nterarrval tmes are eponentally dstrbuted we can set A ln( R ) α Well known, we derved ths generator n the begnnng of the class 7.9

30 7.30 Acceptance-Rejecton Technque: Posson Dstrbuton Substtute the sum by Smplfy by multply by -α, whch reverses the nequalty sgn sum of logs s the log of a product Smplfy by e ln() + > n n R e R α + < ) ln( ) ln( n n R R α α + + > > ln ln ) ln( ) ln( n n n n R R R R α α

31 Acceptance-Rejecton Technque: Posson Dstrbuton Procedure of generatng a Posson random varate N s as follows. Set n0, P. Generate a random number R n+, and replace P by P R n+ 3. If P < ep(-α), then accept Nn Otherwse, reject the current n, ncrease n by one, and return to step. 7.3

32 Acceptance-Rejecton Technque: Posson Dstrbuton Eample: Generate three Posson varates wth mean α0. ep(-0.) Varate Step : Set n 0, P Step : R , P Step 3: Snce P < ep(- 0.), accept N 0 Varate Step : Set n 0, P Step : R 0.446, P Step 3: Snce P < ep(-0.), accept N 0 Varate 3 Step : Set n 0, P Step : R , P Step 3: Snce P > ep(-0.), reject n 0 and return to Step wth n Step : R 0.995, P Step 3: Snce P > ep(-0.), reject n and return to Step wth n Step : R , P Step 3: Snce P < ep(-0.), accept N 7.3

33 Acceptance-Rejecton Technque: Posson Dstrbuton It took fve random numbers to generate three Posson varates In long run, the generaton of Posson varates requres some overhead! N R n+ P Accept/Reject Result P < ep(- α) Accept N P < ep(- α) Accept N P ep(- α) Reject P ep(- α) Reject P < ep(- α) Accept N 7.33

34 Specal Propertes 7.34

35 Specal Propertes Based on features of partcular famly of probablty dstrbutons For eample: Drect Transformaton for normal and lognormal dstrbutons Convoluton 7.35

36 Drect Transformaton Approach for N(0,) PDF f ( ) e π CDF, No closed form avalable t F( ) e π dt 7.36

37 Drect Transformaton Approach for N(0,) Consder two standard normal random varables, Z and Z, plotted as a pont n the plane: In polar coordnates: Z B cos(α) Z B sn(α) Z (Z,Z ) B α Z 7.37

38 Drect Transformaton Ch-square dstrbuton Gven k ndependent N(0, ) random varables,,, k, then the sum s accordng to the Ch-square dstrbuton PDF χ k k f (, k) Γ ( k ) k k e 7.38

39 Drect Transformaton The followng relatonshps are known B Z + Z ~ χ dstrbuton wth degrees of freedom ep(λ /). Hence: B lnr The radus B and angle α are mutually ndependent. Z ln R cos(πr ) Z ln R sn(πr ) 7.39

40 Drect Transformaton Approach for N(µ, σ ): Generate Z ~ N(0,) µ + σ Z Approach for Lognormal(µ,σ ): Generate ~ N(µ,σ ) Y e 7.40

41 Drect Transformaton: Eample Let R and R Two standard normal random varates are generated as follows: Z Z ln(0.758) cos(π 0.489). ln(0.758) sn(π 0.489).50 To obtan normal varates wth mean µ0 and varance σ

42 Convoluton Convoluton The sum of ndependent random varables Can be appled to obtan Erlang varates Bnomal varates 7.4

43 Convoluton Erlang Dstrbuton Erlang random varable wth parameters (k, θ) can be depcted as the sum of k ndependent eponental random varables,,, k each havng mean /(k θ) k k ln kθ ln( R ) kθ k R 7.43

44 Summary Prncples of random-varate generaton va Inverse-transform technque Acceptance-rejecton technque Specal propertes Important for generatng contnuous and dscrete dstrbutons 7.44

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