Chapter Taking The III : distribution 's of view Point is :* : : - PowerPoint PPT Presentation

Chapter Taking The III : distribution 's of view Point

is :* : : : : : " " " distribution can , like a random variable we find a so that X - µ ② methods for creating distributions " distribution " use this ③ How perspective we

A distribution p a distribution ? : 93 - s IR what is - algebra 931 function probability IR ( w/ r on is a of µ information encoded in The is Recall too : defined by Cdf : IR - SIR corresponding Fx the = y l l - o , sis ) Fx ( x ) have ? a cdf properties does what - decreasing ① ¥7 ③ Fx is non - O . . Fxcx ) - continuous right is ④ Fx ,fknHFkD ③ ¥7 lie , if Exam a. the SF - I . Fxlk ) -

maker ) Theorem ( Random variable - ④ of properties ① satisfies F : IR - HR Ippon that a random ravine X en exists There Then chef . particular , it µ a is - Fx a . In F so that - lo , D exists IR , then there a measure for 93 probability an X ya that X variable . random so au lo , D distribution uniform II let . The be U to :( Oil ) → lo , D by define - flu ) We X : FCK ) > u } - claim . we = int { x flu ) = F Fx has .

us FCK ) iff Glu ) s x claim : Big we got continuous , right F is why ? Since = min { x : FCK ) > u } - in f { x : FCK ) tu } flu ) - FCK ) > u so of x the smallest value flu ) is So x > flu ) if F- ( x ) su , Then Hence inequality F is - other non , since For the then have pea ) s x decreasing we u EFC local ) E FCK ) . FCK ) = BLUE FCK ) ) IP ( local Ex ) So : = . DM

Maker ) Core ( Independent Random Variable of probability Anubis be sequence , , µ any let y . . - Xi , Xz , . variable , exist rhndom . - there Then IR on . Xi - mi that independent so . that are e- to produce - o - fact use this how we can So : new random hence ( and distributions new variables ) ? the answers have we'll

distribution ) ( Density begets therm Borel mensurable f : IR - i IR is suppose that a and I Ht ) Hdt ) =L f- 30 satisfies function that . , by : 93 - HR function the Then p a valid probability = fpflt ) 1pct ) Hdt ) is B ) pl a distribution ) ( and hence function . a probability function properties of check The . PI k$1

distributions ? How else create we can ( Adding Distributions ) theorem , and if { pile IR > o distributed Suppose µ , yo , are . . . Then f- ? pi Mi so that Epi - 1 , is a - distribution too . be a : 93 → IR to fer g properties cheek PI . ( use The fnot That Mi obey probability measure axioms . ) confirm desired properties to relevant

for µ statistics for µ :* are statistics related to How if - Z pi Mi ? p - distribution ) linear in theorem ( Expectation is with - Epi µ each by given - . : 93 - HR p is let y mensurable for Borel distribution . Then ay µ a = ? pi Eff ) have Eff ) filth → R , m - - ? pi frflt ) mild t ) fpflt ) Mdt )

- IB cheek this with fer f Be 73 . PI we'll We - usual way " in the result " the general get . ) by limit , etc smipkf ( extend to = ? pi ' Emil Hp ) Epl # B ) need So : . we - MLB ) ' ) . MB - pl B) to IE , this ) = 1 - LHS : - l ? piri ) ( B) = ? pi Mi ( B ) : Epi Emil Hp ) RHS - - { pipe by sides some These - µ agree . . hypothesis DM .

how haiwitg of expectation in distribution play , at ) Ey ( see t I t t ft - I 81 Consider pinion , - p with Xiyu random variable be . X a let chap at variables IE ! X ) ? By : What is - Eyelid ) Ep ( id ( x ) ) IE ( X ) - - - les , lid ) t I term . . . , lid ) = Is Eg , lid ) t t go 's ,{¥it% = Is t 's .tt . I

be sgmnjfxn ' ) : IR Ep ( X ? HR let What sq - . is 1Er ( sq ) Ep ( sq ( x ) ) Ep ( X ' ) = = IES , Csg ) t Is Emma , (9) - t IES , Isg ) t f - - it 't Is § e- t% ' t's ' - t . Hdt ) . t . I - = ¥ i ¥ . I Th . I ' t ¥ . - ( Ep ( X ) ) ' Ep ( X ' ) N ( X ) = then : = Nak t E. it to ) ⇐ t's t 't E ) - ft