Punt: T , T , T are mutually independent . How about T , the - PowerPoint PPT Presentation

Poisson Distribution: Review Poisson Over Time - neg Let B 1 ∼ Poisson( λ ) be the number of bikes that integers non Values: are stolen on campus in one hour. (Go bears?) The Poisson Arrival Process Parameter(s) : X " rate " , What is the distribution of B 2 . 5 , the number of CS 70, Summer 2019 a P [ X = i ] = e- bikes that are stolen on campus in two hours? . it Atd half . a ( 2.52 ) Poisson Bonus Lecture, 8/14/19 Bz E [ X ] = x ~ . as 2.5 X ECB 2.57 Var[ X ] = x = . X Rate over time T = T 1 / 22 2 / 22 3 / 22 Adding Poissons: Review Adding Poissons: Twist? Decomposing Poissons Let T 1 ∼ Poisson( λ 1 ) be the number of particles What is the distribution of the total number of Let T ∼ Poisson( λ ) be the number of particles detected by Machine 1 over 3 hours. particles detected across both machines over 5 detected by a machine over one hour. hours? M1 from in 1 hour Let T 2 ∼ Poisson( λ 2 ) be the number of particles particles Each particle behaves independently of others. ' # T = , detected by Machine 2 over 4 hours. " " M2 " " " ' Tz = Each detected particle is an α -particle with Poisson ( ¥ ) Tin The machines run independently . probability p , and a β -particle otherwise. Tz ' Poisson ( ⇒ - What is the distribution of T 1 + T 2 ? Let T α be the number of α -particles detected by poi ( ¥ -1¥ ) a machine over one hour. What is its distribution? + Tz hour ' ' 1- T , : n + Az ) t Tz ( X , Poisson T , n " poi [5131+7*7] hour - s : 4 / 22 5 / 22 6 / 22

: PETA Decomposing Poissons Goal a ] Independence? Decomposing Poissons Remix - - Yes Let T α be the number of α -particles detected by Are T α and T β independent? Now there are 3 kinds of particles: α , β , γ . . a machine over one hour. What is its distribution? Each detected particle behaves independently of ) ] Pta ]=§=fpa=a)nfT=N a ← Total Prob - t.tn ) ( . others, and is α with probability p , β with n =q7a ( e probability q , and γ otherwise. ( xp ) Poisson T α ∼ " E.ee#e.xa.pE...n;n.naa:aehs.g.iE9tmn-a=fe-xp.ena.a " " " - )¥yjf¥÷EnaIhP ( Aq ) Poisson T β ∼ CF - GD - p Poisson Cali " " T γ ∼ ma " Forgeries - xp HPa fore =e . ¥ Punt: T α , T β , T γ are mutually independent . How about T β , the number of β -particles? Poisson ( X ) Sanity Check: T α + T β + T γ ∼ - PD CHI Poisson Tp Poisson ( xp ) - - 7 / 22 8 / 22 9 / 22 Exponential Distribution: Review Break Poisson Arrival Process Properties We’ll now work with a specific setup: o ) ( O Values: I There are independent “arrivals” over time. , I The time between consecutive arrivals is Parameter(s) : x Expo( λ ) . We call λ the rate . XX " " fx ( x ) Xe - M If you could rename the Poisson RV (or any RV PDF P [ X = i ] = - : Times between arrivals also independent . - for that matter), what would you call it? I For a time period of length t , the number of ÷ arrivals in that period is Poisson( λ t ) . E [ X ] = I Disjoint time intervals have independent 1- Var[ X ] = numbers of arrivals. 72 10 / 22 11 / 22 12 / 22

Poisson Arrival Process: A Visual Transmitters I Transmitters I txt ) # arrivals Poisson A transmitter sends messages according to a How many messages should I expect to see from - # Poisson Process with hourly rate λ . 12:00-2:00 and 5:00-5:30? I ti t ¥005730 Given that I’ve seen 0 messages at time t , what is 12`00 2G00 t t t the expected time until I see the first? .5h Expo ( X ) 2h to Total Time : X ~ ' , 5h ' 2. p [ xzs.tt/XZtI=lPfxzs - inte ) ness less o memory : Poisson ( 2.5A ) in arrivals in # - " NEXPOCX ) " reset At intervals time t can both , Intuition : 't time in t as time O time ]=¥ Treat time ⇒ Ef# messages ) Eflnter arrival . 2.57 # - - at Efpoisggon after t arrival time a- Expected first = unit see time - ⇒ arrival ← inter - 13 / 22 14 / 22 15 / 22 Transmitters II: Superposition Transmitters II: Superposition Transmitters II: Superposition Transmitters A, B sends messages according to Transmitters A, B sends messages according to If the messages from A all have 3 words, and the I Poisson Processes of rates λ A , λ B respectively. Poisson Processes of rates λ A , λ B respectively. messages from B all have 2 words, how many The two transmitters are independent . The two transmitters are independent . words do we expect to see from 12:00-2:00? from A , MA # messages 12G00 2G00 = - We receive messages from both A and B . We receive messages from both A and B . " " B ' ' " MB = , #X# A XA . 2) ÷H#¥H¥ eat Poisson ( XA MA - - 2) ( X B Poisson MB ~ :* . What is the expected amount of time until the What is the expected amount of time until the Ef words I 2 MBT E- [ 3 MAT = first message from either transmitter? first message from either transmitter? 62 At 4 XB - ¥tTB 2 ECMB ] 3 ECMA It - = Expo ( XATXB ) ECT ] - T - - 16 / 22 17 / 22 18 / 22

Kidney Donation: Decomposition Kidney Donation: Decomposition Kidney Donation: Decomposition My probability instructor’s favorite example... If I have blood type B, how long do I need to wait Now imagine kidneys are types A, B, O with before receiving a compatible kidney? probabilities p , q , ( 1 − p − q ) , respectively. - p ) Ali Kidney donations at a hospital follow a Poisson process B Poisson rate Type : Process of rate λ per day. Each kidney either If I have type B blood, I can receive both B and O. TN Expo ( XG - p ) EGF F- until first B ) time . comes from blood type A or blood type B , with How many compatible kidneys do I expect to see Say I just received a type A kidney. probabilities p and ( 1 − p ) respectively. over the next 3 days? The patient receiving a type A kidney after me is XXXI expected to live 50 more days without a kidney " All # donation. What is the probability they survive? ÷H¥H÷ : kidney A until next T time . = . - q ) xp - p Xu Expo ( Xp ) T - - APX 50 ) - Xpxdx " - e { go.tt#Bokidn..eys.?days-PoiC3aq)EE3If7p PETE 501=150 ape , I = - op ) poi ( 37ft - p ~ 19 / 22 20 / 22 21 / 22 Summary When working with time , use Expo( λ ) RVs. When working with counts , use Poisson( λ ) RVs. Superposition: combine independent Poisson Processes, add their rates. Decomposition: break Poisson Process with rate λ down into rates p 1 λ , p 2 λ , and so on, where p i ’s are probabilities. 22 / 22