tYff distribution of random a variable connection between with - PowerPoint PPT Presentation

Chapter of Distribution I : Random Variable a

Lattin Defy Distributed ) ( Identically variables { Xilie # random of collection A distributed it for all and seek identically is have i. je I we any - IPCX ; ex ) IP ( Xi ex ) . K Identically distributed implies equal integrals ) theorem fer ay identically distributed iff { Xilie # is CECHXI ) ) - IEHCX ; ) ) have f : IR -1112 measurable - we c- I all i. j for .

tYff ① distribution of random a variable ② connection between with Vanities random and d. striations equal " identically distributed "

Dein ( Distribution ) Cr , F , IP ) random variable on be X let a " the law of X " ) ( aka distribution of X the : 13-7112 by given fuckin is the µ Borel sets - ' ( B ) ) IP ( X EB ) =P ( X put B) = . ddkuhh.eu X has X - µ indicate that We write to µ .

AHemaknotnh.is#M--Mx--LCx)--lPX " t ! also notation for IECX ) Danger , aka off ) function distribution Def ( cumulative cdf the , then variable a random If X is ' IR defined by : R - function Fx the of is X - all - a. xD IP ( X a- x ) Fx ( x ) - - - .

are identically Y X and say that to Nate : as saying - Fy Ex distributed the - same is . ( doppelganger ) lemme ② iff = Fu Fx distributed identically , Y ① X and are if f- ③ LIX ) =L ( Y ) . we've already mentioned above PI ① ⇐ ② . midst of the past that in the ① ⇐ ③ given integrals distributed equal implies identically .

distribution implies equal integrals ) Cory ( Equal with variables random Y and If are X if f : IR - Boel HR LIX ) =L ( Y ) , is and a = Eff ( YD ⇐ ( f- ( x ) ) function , Then measurable . distributions result , hairy equal PI last By the identically distributed Y and X . implies are TX

observation : if final , m ) 1112,93 One , then X-p probability space . is a M ) ( R , , m ) 1112,93 ( r ,F , P ) measurable function have Borel we f :D -2112 For a , 1112,93 , µ ) ② HX ) on ( AFP ) variables ① f z random on HOW RELATED ? THEY ARE

Lemmy ( r , F , p ) with variable random If X on is a measurable Borel is f. IRAN if and a X - µ , random variables f : IR → IR the function , then IP ) ( under have - b - ' IR FIX ) and ( under y ) - - Fffx ) Ff - - IP ( f ( X ) EX ) . Fux , ( x ) have let KEIR - PI We - ' ( f - ' ( C . - six ] ) ) ) ( C - A ,x3 ) ) =p ( X - ' =p ( ( full ' ( f Ff ( x ) - A ,x3 ) ) = µ ( f- 1%1 = .

variables ) Cor ( change of before - have setup with the as , we same E- ( f ) IE ( FIX ) ) = KEE p TEK - INI - Lebesgue ) ( not under denoted { flxldlp Hxllw ) Mdw ) IEFHX ) ) Recall : is = § t da = { fft ) Mdt ) HE (f)