Functions of Two Random Variables
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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 P w and w F hn and ar indpndnt, F ( w) F ( w) F ( w),, ( w, w) f ( w) f ( w) F ( w) + f ( w) F ( w) Intuitivl Corrct? Exampl Suppos and ar indpndnt xponntial random variabls both with arrival rat Thn x x f x, x 0 F x, x 0 f, 0 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 P w or w hn and ar indpndnt, F ( w) + F ( w) F ( w, w), + F w F w F w F w F w + f w f w f w f w F w F w f w f w { F w } + f w { F w } Intuitivl Corrct? Exampl Suppos and ar indpndnt xponntial random variabls both with arrival rat x x f x, x 0 F x, x 0 Thn f ( w) f w { F w } + f w { F w } w w w Not that 0 ( ) ~ Exp wf ( w) dw
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, 0 f x x F x x, 0, 0 f F Thn f ( w) f w { F w } + f w { F w } w w w w ( ) ( + ) Not f ( w) ~ Exp + + ( + ) w Exampl Min of Indpndnt Exponntial Suppos, and 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 ( w) Find F 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, ) Exampl Diffrnc btwn Indpndnt Exponntial ( ) ( ) Suppos Exp, Exp, and ar indpndnt Dfin Find th pdf of Solution Sinc Exp, Exp, and ar indpndnt, x, f x x, for, 0, x+ w + w 0 x, + 0, F ( w) dx d f x, d dx f x, can show w w + + F ( w), w 0 d f( w) F( w) dw +, 0 w w + + w f Intuitiv? 4
5 5 ( f ) sas [ ] [ ] f ( w) P f ( w) + P f ( w) w 0 showing th mmorlss proprt of th xponntial distribution Not whn, w f ( w) Homwork First Arrival and ar indpndnt xponntial random variabls with arrival rats and rspctivl [ ] Show P and P[ ] + + 5
6 6 Proprtis: 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 ( ) ( ) + Exp + Exp 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 P + z Z x+ z hn and ar indpndnt, dx d f ( x, ), fz( z) FZ( z) z Using Libnitz's rul, x+ z dx f ( x) d f ( ) x z dx f ( x) d f( ) z + f ( x) f ( x+ z) dx 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 Not: Lt,, U 3 Assum, and Dfin + Find th pdf of 0, ar indpndnt Considr hat will th pdf of look lik? l 8
9 9 Exampl -Erlang Distribution and ar indpndntxponntial 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 f Z ( z) ( z)! 0 z z 0 othrwis Z is rfrrd to as a -Erlang rv Z 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! Δz z 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 Answr: 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 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 Answr: 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 +
Functions of Two Random Variables
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 (
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