# Density Curve. Continuous Distributions. Continuous Distribution. Density Curve. Meaning of Area Under Curve. Meaning of Area Under Curve

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1 Continuous Distributions Rndom Vribles of the Continuous Tye Density Curve Perent Density funtion f () f() A smooth urve tht fit the distribution Test sores Density Curve Perent Probbility Density Funtion f () Shows the robbility density for ll vlues of. f() A smooth urve tht fit the distribution Probbility density is not robbility! Use mthemtil model to desribe the vrible. Test sores Continuous Distribution Probbility Density Funtion (.d.f.) of rndom vrible X of ontinuous tye with se S is n integrble funtion f() tht stisfying the following onditions:. f () S. S f() d = (Totl re under urve is.) b. For nd b in S P ( X b) f ( ) d f() b Mening of Are Under Curve Emle: Wht erentge of the distribution is in between 7 nd 6? f() P(7 X 6) Mening of Are Under Curve Emle: Wht erentge of the distribution is in between 7 nd 6? f() P(7 X 6) = P(7 < X < 6) = P(7 X <6) = P(7 < X 6) P(X = 7) = 7 6 X (Height) X (Height) density is not robbility

2 Emle: If the density funtion of ontinuous distribution is for. f ( ) elsewhere Find the roortion of vlues in this distribution tht is less thn /. / The re under the f () between nd / = d / f () = y / = (/) () = /6 = /.. f () = is line with sloe nd sses through () 7 n d n Review of Clulus n u e d e du u e d u du d e d e Emle: If the density funtion of ontinuous distribution X witing time between rrivls of rs t intersetion is f ( ) e for Find the robbility tht the witing time (in seonds) till the net rrivl of r t this intersetion is more thn seonds. - The re under the f () nd > = e d b / lim ( e / / ) lim( e b ) ( e ) (.). b b f () = y. Emle: If the density funtion of ontinuous distribution X witing time between rrivls of rs t intersetion is f ( ) e for Find the robbility tht the witing time (in seonds) till the net rrivl of r t this intersetion is less thn seonds. - The re under the f () below = e d / ( e / / / ) ( e ) ( e ) e... f () = y 9 Cumultive Distribution Funtion The umultive distribution funtion (.d.f. or distribution funtion d.f.) of ontinuous rndom vrible is defined s F ( ) P( X ) f ( t) dt F( ) = F() = P( < X < b) = F(b) F() F () = f() if derivtive eists Emle: If the.d.f. of ontinuous distribution X witing time between rrivls of rs t intersetion is where is onstnt rmeter of this distribution f ( ) e for Find the distribution funtion. F ( ) ( e F ) e f ( t) dt t - e t / / ) ( / e dt

3 Mesure of Center for Continuous Distribution The men vlue (eeted vlue) of ontinuous rndom vrible (distribution) X denoted by X or just (or E[X]) is defined s X f ( ) d Mesure of Sred for Continuous Distribution The vrine of ontinuous rndom vrible (distribution) X denoted by X or just (or Vr[X]) is defined s E[( X ) ] ( ) f ( ) d The stndrd devition of X is E[( X ) ] Moment Generting Funtion for Continuous Distribution The moment generting funtion if eists of ontinuous rndom vrible (distribution) X denoted by M(t) is defined s E[ e tx ] e t f ( ) d h t h Emle: If the density funtion of ontinuous distribution is for. f ( ) elsewhere Find the men nd vrine of this distribution The men is. E( X ) f ( ) d f ( ) d d.. (. ) 6 Emle: If the density funtion of ontinuous distribution is for. f ( ) elsewhere Find the men nd vrine of this distribution The vrine is Vr ( X ) E[ X ] ( E[ X ]) The Perentile The ()th erentile (quntile of order ) is the number suh tht the re under f() to the left of is. E[ X ] E[ X. f ( ) d. d f ( ) d. ] ( E[ X ]) (.6 ) 7 ( ) d F( ) M () M () 7 f Medin is the th erentile.

4 Medin of distribution Emle: If the density funtion of ontinuous distribution is f ( ). elsewhere Find the medin of this distribution. f () = y for Medin is suh tht Wht is? d.. /. d... 9 Perentile Emle: If the density funtion of ontinuous distribution is f ( ) for. elsewhere Find the th erentile of this distribution. f () = y th erentile is suh tht Wht is? d.. / 6. d... Smle Quntile Let y y... y n be the order sttistis ssoited with the smle n then y r is lled the smle quntile of order r r s well s erentile. n Emle: 7 9 n n = 6 nd the vlue y Other otions: r. n is the [/(6+)]th quntile of the distribution of the smle i.e..7 smle quntile or 7. erentile. r.7 n. Emine Distribution with Quntile-Quntile Plot Theoretil Distribution Smle y r Good model for the observtions Theoretil Distribution * Mke lot using dt in revious slide. Smle y r Not good model for the observtions

5 Continuous Distributions The Uniform nd Eonentil Distributions Uniform Distribution The ontinuous rndom vrible X hs uniform distribution if its.d.f. is equl to onstnt on its suort. If the suort is the intervl [ b] then its.d.f. is f ( ) b. b It is usully denoted s U( b). b f() F() b b 6 Uniform Distribution The men vrine nd m.g.f. of ontinuous rndom vrible X tht hs uniform distribution re: b tb t e e t( b ) t t. ( b ) Pseudo-Rndom Number Genertor on most omuters U( ) 7 Uniform Distribution Emle: Let X be U() find the men nd the vrine nd the m.g.f. of X. t e t ( ) t t. Eonentil Distribution The ontinuous rndom vrible X hs n eonentil distribution if its.d.f. is f ( ) / for elsewhere where is the men of the distribution. * X n be the witing time until net suess in Poisson roess. 9 Eonentil Distribution The men vrine nd m.g.f. of ontinuous rndom vrible X tht hs n eonentil distribution re: t.d.f. F ) e t ( /

6 Eonentil Distribution Emle: Let X hve n eonentil distribution with men of wht is the first qurtile of this distribution? F( ) / e /./ ( ). F e. e.7.. / = ln(.7). = ln(.7) { = }. = ln(.7). =.6 Eonentil Distribution Let W be the witing time until net suess in Poisson roess in whih the verge number of suess in unit intervl is l then for w F(w) = P(W w) = P(W > w) = P(W > w) = P(no suess in [w]) = e -lw d.f. of eonentil distribution. F (w) = f(w) = le -lw.d.f. of eonentil distribution. / = ln( ) e Eonentil Distribution Let W be the witing time until net suess in Poisson roess in whih the verge number of suess in unit intervl is l then l = / where is the verge witing time until net suess for w F(w) = e -lw = e -w/ f(w) = w/ e d.f. of eonentil distribution..d.f. of eonentil distribution. Eonentil Distribution Suose tht number of rrivls of ustomers follows Poisson roess with men of er hour. Wht is the robbility tht the net ustomer will rrive within minutes? ( min. =. hour) P(X ) = F () = e -/ P(X.) = F (.) = e -./ l = =. F (.) = e -./. =.9 Gmm Distribution Continuous Distributions The Gmm nd Chi-squre Distributions The ontinuous rndom vrible X hs Gmm distribution Gmm( ). if its.d.f. is / e for f ( ) ( ) elsewhere. Gmm Funtion: (n) = (n )! * X n be the witing time until th suess in Poisson roess. 6 6

7 Grhs of Gmm Distributions = / = = = = = = = /6 = = = = 7 Gmm Distribution The men vrine nd m.g.f. of ontinuous rndom vrible X tht hs n Gmm distribution re: ( t) t Gmm( ) Eonentil Distribution Gmm( r/ ) Ch-squre with d.f. = r. Seil Nottion (r) Let X be rndom tht hs Chi-squre distribution with degrees of freedom r P[ X ( r)] Til re Probbility of X greter thn or equl to (r) is. P[ X ( )] r (r) Emle: Find.() =? 7. 9 Try This! Find.(7) =? Find.() =? Find the th erentile from distribution with degrees of freedom 6. Continuous Distributions Distributions of Funtions of Rndom Vrible - Rndom number genertion 7

8 A Good Genertor X i = (979 X i- ) mod ( ) Strts with Seed number Simle Widesred use Long yle length (ll number besides nd n be generted.) U i = X i /( ) U i ~ U() Methods for Generting Non-Uniform RN s CDF Inversion Trnsformtions Aet/Rejet Methods CDF Inversion Theorem: Let U hve distribution tht is U(). Let F() hve the roerties of distribution funtion of the ontinuous tye with F() = nd F(b) = nd suose tht F() is stritly inresing on the suort < < b where nd b ould be nd resetively. Then the rndom vrible X defined by X = F (U) is ontinuous rndom vrible with distribution funtion F(). Proof: Let X = F (U) the distribution funtion of X is P (X ) = P [ F (U) ] < < b. Sine F () is stritly inresing { F (U) } is equivlent to { U F () } nd hene P ( X ) = P [ U F () ] < < b. But U is U( ); so P ( U u ) = u for < u < nd ordingly P ( X ) = P [ U F () ] = F () F() <. Tht is the distribution funtion of X is F(). 6 Generte rndom numbers from Eonentil distribution = F() = e / nd =F (y) = ln( y) Use uniform U( ) rndom number genertor to generte rndom numbers y y... y n. The eonentilly distributed rndom numbers i 's would be i = ln( y i ) i =... n. Therefore if the uniform rndom number genertor genertes number. then.67 = ln(.) would be rndom observtion from eonentil distribution with =. 7 Try this!!! An U() rndom number genertor hs generted vlue of.6.. Use the CDF Inversion method to onvert this U() rndom number to simulte n observtion from n eonentil distribution with =. i = F (y i ) = ln( y i ). Use CDF Inversion method to generte rndom number from the ontinuous rndom vrible tht hs the following.d.f. for. f ( ) elsewhere

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