Functions of Two Random Variables

Transcription

1 Functions of Two Random Variabls Maximum ( ) Dfin max, Find th probabilit distributions of Solution: For an pair of random variabls and, [ ] F ( w) P w [ and ] P w w F, ( w, w) hn and ar indpndnt, F ( w) F ( w) F ( w), f ( w) f ( w) F ( w) + f ( w) F ( w) m Can ou vrif q m intuitivl? Exampl Suppos and ar indpndnt xponntial random variabls both with arrival rat Thn x x f x F x, x 0 f F, 0 f ( w) f w F w + f w F w w w ( ) w w, w 0 Not that 3 wf ( w) dw 0

2 Minimum ( ) Dfin min, Find th probabilit distributions of Solution: For an pair of random variabls and, [ ] F ( w) P w [ or ] P w w F ( w) + F ( w) F ( w, w), hn and ar indpndnt, + F w F w F w F w F w + f w f w f w f w F w F w f w Can ou vrif q m intuitivl? { } { } ( ) f w F w + f w F w m Exampl Suppos and ar indpndnt xponntial random variabls both with arrival rat x x f x F x x 0 Thn f ( w) f w { F w } + f w { F w } w w w ( m ) 3 q m3 sas ( ) ~ Exp

3 3 Exampl Min of Indpndnt Exponntial Suppos and ar indpndnt xponntial random variabls with arrival rat and rspctivl Dfin min, Find th pdf of Solution: x x 0 f x F x x 0 f F Thn f ( w) f w { F w } + f w { F w } Eq m4 sas w w w w + ( + ) w + w 0 m4 ( ) ( + ) ~ Exp Exampl Min of Indpndnt Exponntial Suppos, and Z ar indpndnt xponntial random variabls with arrival rats, and rspctivl x z ( Z) Dfin min,, Find th probabilit distributions of Solution: ( x + + z) ~ Exp ( ) ( x z) f ( w) + + w 0 x z + + w 3

4 4 Diffrnc Assum, 0 Dfin Find th distributions of For w 0, F ( w) P w [ ] P w + w shadd ara f, ( x, ) dxd x+ w + w 0 x, 0, dx d f ( x, ) + d dx f ( x, ) d Exampl Diffrnc btwn Indpndnt Exponntial ( ) ( ) Suppos Exp, Exp, and ar indpndnt Dfin Find th pdf of Solution ( ) ( ) Sinc Exp, Exp, and ar indpndnt, From q d, can show x, f x x,, 0 x+ w + w 0 x + 0,, F ( w) dx d f x, d dx f x, w w + + F ( w) w 0 d f( w) F( w) dw + w 0 d w w + + Can ou vrif q d intuitivl? 4

5 5 Homwork Th First Arrival and ar indpndnt xponntial random variabls with arrival rats and rspctivl [ ] Show P and P[ ] + + Eq d sas [ ] [ ] f ( w) P f ( w) + P f ( w) w 0 showing th mmorlss proprt of th xponntial distribution Not whn, w f ( w) 5

6 6 Rviw: For an pair of random variabls and, max + max min diff min diff f ( x ) f ( ) min ( x) ( ) hn Exp, Exp, and ar indpndnt, Exp ( x + ) Exp Exp ( ) + ( ) x diff x x + x + hn, x min diff max Exp Exp min ( ) ( ) + diff max is not xponntial, is a sum of xponntial random variabls 6

7 7 Sum of Indpndnt Continuous Random Variabls Lt Z + If and ar indpndnt, f ( z) f ( z) f ( z) convolution Z f ( τ) f ( z τ) dτ f ( z τ) f ( τ) dτ proof : [ ] F ( z) P + z Z [ ] P + z x+ z dx d f ( x, ) hn and ar indpndnt,, x+ z Z F ( z) dx f ( x) d f ( ) fz( z) FZ( z) z using Libnitz's rul x z dx f ( x) d f( ) z dx f ( x) f ( x + z) f ( x) f ( z x) dx + 7

8 8 Exampl, U ( 0, ) and ar indpndnt Dfin Z + Find th pdf of Z fz ( z ) f( x) f( ) f ( τ ) f ( z τ) dτ 0 z < 0 z 0 z < z z < 0 z Exampl Lt,, U 3 Assum, and Dfin + Plot th pdf of 0, ar indpndnt Considr n hat will th pdf of look lik as n? 8

9 9 Exampl -Erlang Distribution and ar indpndnt xponntial rvs with arrival rat Lt Z + f ( z) f ( z) f ( z) Z For z 0, f ( τ) f ( z τ) dτ z 0 τ ( z τ ) f ( z) dτ Z z z z ( z) f z 0 Z z! 0 othrwis Z Suppos buss arriv in a Poisson arrival procss with rat Lt b th waiting tim until th nxt bus, and lt b th intr-arrival tim of th following bus ( ) Thn and ar indpndnt Exp Z + is th waiting tim until th scond bus Z is rfrrd to as a -Erlang rv Not From q,, ( z) z P[ z Z z+δ z] fz ( z) Δ z Δz! 9

10 0 Homwork m-erlang Distribution Z m, whr i ar indpndnt xponntial rvs with arrival rat Show 0 z < 0 f m z Z z ( z) z 0 ( m! ) Z is rfrrd to as a m-erlang random variabl Not Z is th tim of th m m ( z) ( m ) th arrival th fz ( z) Δ z Pm arrival occurs in tim z, z+δz Pm arrivals occur in tim 0, z Pon arrival occurs in tim z, z+δz z! Δz Not Th Gamma-distribution is a gnralizd vrsion of th Erlang distribution α x ( x) f ( x) x 0 with α > 0, > 0 Γ ( α) α is an positiv ral numbr 0

11 Homwork Sum of Indpndnt Gaussian x x and ar indpndnt Gaussian rvs with man μ and μ, and varianc σ and σ rspctivl Lt Z + Find th pdf of Z Hint f ( z) f ( z) f ( z) Z π ( σx + σ) ( μx μ, σx σ) N + + ( z μx μ) ( σx + σ) Not : Sum of indpndnt Gaussian is Gaussian Th man is th sum of th mans Th varianc is th sum of th variancs

12 Convolution btwn Discrt Distributions Lt Z + hn and ar indpndnt intgr random variabls, th pdf of Z is th convolution of th pdf of and p ( k) p ( k) p ( k) Z p ( j) p ( k j) j Homwork Sum of Indpndnt Poisson and ar indpndnt Poisson random variabls with arrival rats and rspctivl Lt Z + Find th pmf of Z Hint k k and p k p k for k 0,,, k! k! p ( k) p ( k) p ( k) Z ( + ) k ( ) + k 0,,, k! Not Sum of indpndnt Poisson is Poisson Z is a Poisson random variabl with arrival rat +