Chapter & Discrete I : continuous Random Variables
last variable X :D - sik , random For distribution by µ :B - sik get a • - ' ( B ) ) LPC X IPLXEB ) MB ) - - = • get HR kg : IR cdf Fx - a C - A. x ] ) = µ IP ( X E x ) ( Fx ( x ) = 1Er ( f ) "changeofvavibCe HE ( f- ( X ) ) ° =
NawSHffWT " ① Describe basic " some in order variables random discrete characterize to variables random to characterize ② Gie way a variables random continuous distributions their via version et Law ③ Continuous Statistician of lazy
PlX=c ) - I I CHR Suppose - let . . Then if BeBe and Then if Xnm , - { to - ' ( Bl ) CEB IP ( x µ (B) - = CEB - I , (c) - ( Point - mass distribution ) Defy - Ipcc ? 8dB ) : 93 → IR by - Sc define For ER ,
measurable f : IR - YR , we Borel have For Observe any : = Eplf Ix ) ) = fk ) = IE ,p(fk ) ) ( f ) = IES . § Htt Sddt ) "ch¥arib " , random discrete we defined before that from know we took which variables a random on variables as - pi IP ( X valves 443 , with - xi ) - number of countable - satisfying ? pi - I - .
" all the probability definition only that " insists Our new number et a countable bound in for up X is values . Variable ) Random Defy ( Discrete discrete it then called X is variable random A so that c- IR a countable collection x. , xz , it . . . with ? pi particular , - I . In - pi - ki ) IP ( X - - - - 5) = O IP ( X c- Ex . ,xz , have . . . we
. v . ) chunotcnzahin of discrete r Then ( Distribution distribution µ with variable X random A and . EIR if exist there x.pk discrete , . . is pi Sci = E That - I µ . so Epi - observing from direction comes " " ⇒ PI The = § out - ' ( B ) ) = Pl X jB LB ) Pj µ = ? = Epi 8 , CB ) pi ftp.lki ) . ④ ( you do the other half )
continuous variables ? random about What ( Nl 0,11 ) Exe " definition " old was . The X - NCO , l ) Suppose = fo " - t% at Fx ( x ) '¥ e 93 → IR ? distribution is the . n what . iiii÷iii7 " " i' iii. " . irani s -
( Absolutely random ) variables Continuous Define absolutely called X variable is random A X ) exists som if respect to the ( with centners distribution µ the that - HR filth faction so Borel by given of X is = fr Ht ) Holt ) Hdt ) µ (B) . function for density called the f function is The µ .
- Sc that claim Non X Suppose we . need to show absolutely continuous . we not X is measurable f : IR -1112 that so ' no There is that Htt 1pct ) Hdt ) § £ ( B ) . = : Sella ) , I particular = - Hplc ) . In . Sc CB ) Nele : - that Observe = Hc ) HKD to HMH ) - o { Ht ) Health Htt ) - if t Hc ) - c { - if the o
statistician ) law of lazy of thou ( continuous version with density f continuous is X yr . that suppose : IR -7112 , we get measurable Borel g for Then any = § gltsflt ) Mdt ) Et fr GH ) Mdt ) ' IEM g) Tch any variables of I Eplgcx ) )
functions duck this for indicate Pf we 'll again . lie , function for simple functions follows , Then It expectation , then lcwmty A image ) by with finite MCT , and then by - negation Anakin , fer nen - g- - gt by functions gaunt g - for . Beret set . - Hp B for a let So : g - Then
= ! I , It ) on Cdt ) 1Er ( HB ) - pl - pl ' ) B B ) t O = I ( B ) = µ = ! n ) ' [ def f- It ) ht , It ) Ndt ) Daa
of expected value ) - fashioned defa for ( old with density f continuous XY . is Suppose = fptflt ) Hdt ) Epl X ) Then . Ep ( idk ) ) HI ( X ) IE = " ) = Eyelid ) [ chge of variables " ( last result ) Ip fit ) idk ) Htt ) = T ⑤
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