SIO 22B, Rudnick dpted fom Dvis III. Single vile sttistics The next few lectues e intended s eview of fundmentl sttistics. The gol is to hve us ll speking the sme lnguge s we move to moe dvnced topics. This my e too elementy fo some students, ut thee is possiility tht the pesenttion will e fom slightly diffeent pespective thn you hve seen efoe. Fist, we need some definitions. A ndom vile is vile whose vlue is detemined y ndom pocess. If pocess poduces vlues tht e not pefectly pedictle fom wht is known, it is ndom pocess. A pooly contolled expeiment might e consideed ndom pocess Any function of ndom vile is lso ndom vile. A eliztion of the pocess poduces one ndom vlue of the vile. A lge collection of eliztions poduced unde sttisticlly identicl conditions (the sme deteministic pmetes) is n ensemle of identiclly peped osevtions. The ct of flipping coin is ndom pocess. If we let x = fo heds nd - fo tils, x is ndom vile. A single flip poduces eliztion of x nd n ftenoon of flipping using the sme technique would yield n ensemle of eliztions. x is A centl concept is the vege o expected vlue of ndom vile. The vege of x = lim x n () n= whee xn is the vlue of x in the nth eliztion. The nottion will hencefoth e eseved fo the idel ensemle vege equiing n infinite nume of eliztions. A complete desciption of ndom vile is given y its cumultive distiution function ( ) = Poility tht x (2) Poility my e thought of s the fction of eliztions. (Figue ) Tke note of the pehps unusul nottion tht hs the ndom vile x s suscipt nd the distiution function depending on the deteministic vile. Some popeties of the distiution function e ( ) ( s) if s (3) ( ) = (4) ( ) = (5) The poility density function (pdf) is ( ) = d d ( ) (6) so tht ( )d = Poility tht < x + d (7)
SIO 22B, Rudnick dpted fom Dvis 2.9.8.7.6 ().5.4.3.2. 4 3 2 2 3 4 Figue. A noml cumultive distiution function..4.35.3.25 ().2.5..5 4 3 2 2 3 4 Figue 2. A noml poility density function.
SIO 22B, Rudnick dpted fom Dvis 3 (Figue 2) Some popeties of the pdf e ( ) (8) The pdf is the diffeentil limit of histogm. ( )d = (9) ( ) = ( s)ds () An inteesting epesenttion of the pdf is s the vege of the delt function Recll tht fo < ( ) = δ ( x) () if <c< δ ( c)d = (2) othewise To undestnd this, conside the distiution function nd let e lge (effectively infinite nume of eliztions) nd M e the nume of eliztions less thn. ( ) = M = δ ( s x n )ds (3) n= Hee the sum is just n unusul wy of witing M. ote tht the expession ove hs the sme fom s the vege (), so tht nd using () the esult () follows. The pdf immeditely gives the vege of ny function g x ( ) = δ ( s x) ds (4) ( ) = g = g ( )δ ( x)d ( ) δ ( x) d = g( ) ( )d In the second step ove we chnged the ode of the integtion nd the vege, nd moved the function g( ) out of the vege. This is possile ecuse the vege is line opeto pplied only to the ndom vile x. Fo exmple the vege o men vlue of x is (5) x = ( )d (6)
SIO 22B, Rudnick dpted fom Dvis 4 Given the pdf of ndom vile x we cn lso find the pdf of ny othe ndom vile y( x) tht is function of x. The compnion deteministic viles nd s e then elted y the sme function s = s( ). Conside fist tht this function is monotonic, then y (7) ( s) = (Figue 3) whee the solute vlue tkes ce of monotoniclly incesing o decesing functions. If the function s (Figue 3). If ll n vlues hve equl poility d (7) ( ) is non-monotonic, nd the invese ( s) is n-vlued then the pdf is n ( s) = i (8) i= d i ( s) = n d (9) 7 6 6 4 5 2 4 y y 8 3 6 2 4 2 4 3 2 2 3 4 x 4 3 2 2 3 4 x Figue 3. A monotonic () function nd non-monotonic () function. Moments The pdf is complete desciption of ndom vile, ut it is often not possile to detemine the pdf fom dt. Pcticl dt nlysis often involves simple sttisticl mesues. We hve ledy discusses the men. A fluctution is defined s x = x x (2) The vince is nd the stndd devition is µ 2 = x 2 (2) σ = µ 2 (22)
SIO 22B, Rudnick dpted fom Dvis 5 The vince cn e thought of s the enegy nd the stndd devition s typicl fluctution size. Highe ode moments e µ n = x n (23) Typiclly, only few low-ode moments cn e detemined fom dt. Highe-ode moments e stongly influenced y the tils of distiution. Exmple pdfs A unifomly distiuted vile is eqully likely to tke ny vlue in n intevl ( ) ( ) fo = othewise Such vile hs men of nd vince of x = d = + 2 (24) (25) µ 2 = 2( ) 2 (26) Mny el-wold ndom viles, especilly those tht e the sum of othe ndom viles, tke on the noml o Gussin distiution ( ) = σ 2π exp ( x )2 2σ 2 (27) with men of x nd stndd devition of σ. The centl limit theoem, discussed lte, will pove why the sum of mny ndom viles poduces nomlly distiuted vile. Pdf demo A pdf is clculted pcticlly s scled histogm. One needs to choose the size nd loction of ins in the histogm. The demo is to show the consequences of this choice.