Special Random Variables: Part 1

Size: px

Start display at page:

Download "Special Random Variables: Part 1"

Julie Richards
5 years ago
Views:

1 Spcl Rndom Vrbls: Prt Dscrt Rndom Vrbls Brnoull Rndom Vrbl (wth prmtr p) Th rndom vrbl x dnots th succss from trl. Th probblty mss functon of th rndom vrbl X s gvn by p X () p X () p p ( E[X ]p Th momnt gnrtng functon s p)+ p p Vr(X) E[X ] (E[X]) p p p( p) (t) E[ tx ] t ( p)+ t p p + p t Chck: (t) p t, (t) p t. () p, E[X ] () p. Bnoml Rndom Vrbl (wth prmtrs n nd p) Th probblty mss functon of th rndom vrbl X s gvn by n p X () p ( p) n,,,...,n

2 p X () n p ( p) n n! (n )!! p ( p) n n! (n )!( )! p ( p) n (n )! np (n )!( )! p ( p) n n X (n )! np (n k)!k! pk ( p) n k k Xn n np p k ( p) n k k k np[p +( p)] n np Altrntly, Bnoml rndom vrbl s numbr of succsss n n trls. X P n X whr X r ndpndnt nd dntclly dstrbutd Brnoull rndom vrbl. " # E X E[X ]np Vr(X) Vr( Th momnt gnrtng functon X ) (t) E[ tx ]E Vr(X )np( p) " xp t!# X ny E [xp(tx )] ( p + p t ) n Chck: (t) n(p t + p) n p t, (t) n(n )(p t + p) n (p t ) +n(p t + p) n p t. Ths gvs () np nd E[X ] () n(n )p + np

3 Gomtrc Rndom Vrbl (wth prmtr p) Lt X dnot th numbr of trls untl succss occurs. Th probblty mss functon of X s gvn by p X () p( p) p p( p) q whr q p p d dq p d dq! q p ( q) /p q q

4 Now, consdr th rndom vrbl Y X,.., E[Y ]E[X] /p. E[XY ] p ( )p( p) ( )q whr q p p d dq! ( )q X q! p d (k )q k dq k!! p d q d q k dq dq k!! p d q d q k dq dq k p d q d dq dq q q p d dq ( q) q p [Bcksubsttut q ( (q ) 3 ( p) p (( p) ) 3 +p p p p p 3 p)] Th momnt gnrtng functon E[XY ]E[X(X )] E[X X]E[X ] E[X] Vr(X)+(E[X]) E[X] ) p Vr(X)+/p /p p Vr(X) p (t) p p t ( p) t

5 Posson Rndom Vrbl (wth prmtr ) Th probblty mss functon of X s gvn by p X ()! ( )! ( )! k [whr k ] (k)! k {z } E[X ]E[X(X ) + X] E[X(X )] + E[X(X )] +. E[X(X ( ) )]! ( )! ( )! k [whr k ] k! k {z } Vr(X) E[X ] (E[X]) E[X(X )] + +. Th momnt gnrtng functon (t) E[ tx ] t!! ( t ) t (t )! Chck: (t) t (t ), (t) ( t ) (t ) + t (t ).Thsgvs () nd E[X ] () +

6 Posson thorm: If n!nd p! suchthtnp! thn n p q n! n!! Ths shows tht for lrg n nd smll p w cn pproxmt th bnoml dstrbuton wth Posson dstrbuton. Contnuous Rndom Vrbl Unform Rndom Vrbl Lt X s unformly dstrbutd ovr (, b). Th probblty dnsty functon s gvn by f X (x), for <x<b b,othrws Z xf X (x)dx Z b b b + b x b dx Z b xdx ppl x b E[X ] Z x f X (x)dx Vr(X) E[X ] (E[X]) (b ). Th momnt gnrtng functon Z b b b (t) tb t(b ) x b dx Z b x dx ppl x 3 b 3 3 ( + b + b ) t

7 Exponntl Rndom Vrbl wth prmtr Th probblty dnsty functon s gvn by Z xf X (x)dx Z x f X (x) x, for x> x,forx< x dx Z x + x x dx [usng ntgrton by prts] E[X ] Z x f X (x)dx Z x x dx [us ntgrton by prts twc] Vr(X) E[X ] (E[X]) Th momnt gnrtng functon (t) E[ tx ] Z tx Z (t Chck: (t) /( t) nd (t) /( t) 3. () / nd E[X ] () /. t x dx )x dx Proprts of th Exponntl Dstrbuton Th xponntl rndom vrbl X s mmorylss,.., P (X >t+ X >t)p (X > ) 8t, Proof: P (X >t+,x > t) P (X >t+ X >t) P (X >t) P (X >t+ ) P (X >t) (t+ ) t P (X > )

8 X nd X r ndpndnt xponntl rndom vrbls wth prmtrs nd, rspctvly. Thn P (X <X ) Z Z Z Z Z Z P (X <X X x) P (X >x) x dx [ P (X ppl x)] x dx [ F X (x)] x dx x x dx + ( + )x dx X,X,...,X n r ndpndnt xponntl dstrbutd rndom vrbls wth prmtrs,,,...,n. x dx P [mn(x,x,...,x n ) >x]p (X >x,x >x,...,x n >x) ny P (X >x) ny [ P (X ppl x)] ny [ ( x )] ny xp Gussn Rndom Vrbl wth prmtrs (µ, ) Th probblty dnsty functon s gvn by " # f X (x) p xp x µ, for <x< " x! x #

9 Assum z x µ. Z xf X (x)dx p Z z + µ) ( z / dz Z ppl Z p z z / dz +µ p z / dz {z } {z } µ Vr(X). Th momnt gnrtng functon of Z (X µ)/ Ths gvs Z(t) E[ tz ] p Z tz z / dz p Z (z tz)/ dz t / p Z (z t) dz {z } t / X(t) E[ tx ]E t( Z+µ) ppl tµ E[ t Z t ]xp + µt ppl (t) (µ + t t )xp + µt ppl ppl (t) (µ + t ) t xp + µt + t xp + µt So, () µ, E[X ] () µ +,Vr(X) E[X ] (E[X]).

10 Dscrt RV Contnuous RV Tbl 4.: Common probblty dstrbutons. Probblty Momnt gnrtng pmf/pdf dstrbuton functon, (t) ) Mn Vrnc Brnoull px() p wth prmtr p px() p p + p t p p( p) Bnoml px() n p ( p) n, wth prmtrs (n, p),,...,n ( p + p t ) n np np( p) Gomtrc px() p( p) pt ( p) wth prmtr p t /p ( p)/p Posson px() wth prmtr (t! Unform fx(x), b tb t ( + b)/ (b ) / t(b ) n th ntrvl [, b] for <x<b Exponntl fx(x) x, t / / wth prmtr for x> h Gussn fx(x) p xp wth prmtrs (µ, ) for <x< x µ, xp h t + µt µ

HW3 : Moment functions Solutions

STAT/MATH 395 A - PROBABILITY II UW Spring Qurter 6 Néhémy Lim HW3 : Moment functions Solutions Problem. Let X be rel-vlued rndom vrible on probbility spce (Ω, A, P) with moment generting function M X.