The Poisson Arrival Process CS 70, Summer 2019 Bonus Lecture, - PowerPoint PPT Presentation

The Poisson Arrival Process CS 70, Summer 2019 Bonus Lecture, 8/14/19 1 / 22

Poisson Distribution: Review - neg integers non Values: X " Parameter(s) : rate " , a P [ X = i ] = e- . it . x E [ X ] = Var[ X ] = x 2 / 22

Poisson Over Time Let B 1 ∼ Poisson( λ ) be the number of bikes that are stolen on campus in one hour. (Go bears?) What is the distribution of B 2 . 5 , the number of bikes that are stolen on campus in two hours? Atd half a ( 2.52 ) Poisson Bz ~ . as 2.5 X ECB 2.57 = . X Rate over time T = T 3 / 22

Adding Poissons: Review Let T 1 ∼ Poisson( λ 1 ) be the number of particles detected by Machine 1 over 3 hours. Let T 2 ∼ Poisson( λ 2 ) be the number of particles detected by Machine 2 over 4 hours. The machines run independently . What is the distribution of T 1 + T 2 ? + Az ) t Tz ( X , Poisson T , n 4 / 22

Adding Poissons: Twist? What is the distribution of the total number of particles detected across both machines over 5 hours? from M1 1 hour in particles ' # T = , M2 " " " " " ' Tz = Poisson ( ¥ ) Tin Tz ' Poisson ( ⇒ - poi ( ¥ -1¥ ) + Tz hour ' ' 1- T , : n " poi [5131+7*7] hour - s : 5 / 22

Decomposing Poissons Let T ∼ Poisson( λ ) be the number of particles detected by a machine over one hour. Each particle behaves independently of others. Each detected particle is an α -particle with probability p , and a β -particle otherwise. Let T α be the number of α -particles detected by a machine over one hour. What is its distribution? 6 / 22

: PETA Goal a ] Decomposing Poissons - - Let T α be the number of α -particles detected by a machine over one hour. What is its distribution? Pta ]=§=fpa=a)nfT=N ) ] ← Prob a Total . - t.tn ) ( n =q7a ( e " E.ee#e.xa.pE...n;n.naa:aehs.g.iE9tmn-a=fe-xp.ena.a " " " - )¥yjf¥÷EnaIhP CF ma " " " Forgeries - xp HPa fore =e . ¥ How about T β , the number of β -particles? - PD CHI Poisson Tp Poisson ( xp ) - - 7 / 22

Independence? Yes Are T α and T β independent? . 8 / 22

Decomposing Poissons Remix Now there are 3 kinds of particles: α , β , γ . Each detected particle behaves independently of others, and is α with probability p , β with probability q , and γ otherwise. ( xp ) Poisson T α ∼ ( Aq ) Poisson T β ∼ - GD - p Poisson Cali T γ ∼ Punt: T α , T β , T γ are mutually independent . Poisson ( X ) Sanity Check: T α + T β + T γ ∼ 9 / 22

Exponential Distribution: Review o ) ( O Values: , Parameter(s) : x XX " - " fx ( x ) Xe M PDF P [ X = i ] = - : - ÷ E [ X ] = 1- Var[ X ] = 72 10 / 22

Break If you could rename the Poisson RV (or any RV for that matter), what would you call it? 11 / 22

Poisson Arrival Process Properties We’ll now work with a specific setup: I There are independent “arrivals” over time. I The time between consecutive arrivals is Expo( λ ) . We call λ the rate . Times between arrivals also independent . I For a time period of length t , the number of arrivals in that period is Poisson( λ t ) . I Disjoint time intervals have independent numbers of arrivals. 12 / 22

Poisson Arrival Process: A Visual txt ) # arrivals Poisson - # ti t t t t ' inte o in NEXPOCX ) Intuition : 't time in time ]=¥ arrival Eflnter # - Efpoisggon a- unit see time - ⇒ arrival ← inter - 13 / 22

Transmitters I A transmitter sends messages according to a Poisson Process with hourly rate λ . Given that I’ve seen 0 messages at time t , what is the expected time until I see the first? Expo ( X ) X ~ , p [ xzs.tt/XZtI=lPfxzs ) less ness memory : " " reset At t time can , as time O t time Treat . at after t time arrival Expected = first 14 / 22

Transmitters I How many messages should I expect to see from 12:00-2:00 and 5:00-5:30? I ¥005730 12`00 2G00 .5h 2h to Total Time : 5h 2. ' - Poisson ( 2.5A ) arrivals in # - intervals both ⇒ Ef# messages ) 2.57 - 15 / 22

Transmitters II: Superposition Transmitters A, B sends messages according to Poisson Processes of rates λ A , λ B respectively. The two transmitters are independent . We receive messages from both A and B . #X# A XA ÷H#¥H¥ :* . What is the expected amount of time until the first message from either transmitter? - ¥tTB ECT ] Expo ( XATXB ) T - - 16 / 22

Transmitters II: Superposition Transmitters A, B sends messages according to I Poisson Processes of rates λ A , λ B respectively. The two transmitters are independent . We receive messages from both A and B . eat What is the expected amount of time until the first message from either transmitter? 17 / 22

Transmitters II: Superposition If the messages from A all have 3 words, and the messages from B all have 2 words, how many words do we expect to see from 12:00-2:00? from A , # messages 2G00 MA 12G00 = - " " ' ' B " MB = , . 2) Poisson ( XA MA - - 2) ( X B Poisson MB ~ Ef words I 2 MBT E- [ 3 MAT = 62 At 4 XB 2 ECMB ] 3 ECMA It - = - 18 / 22

Kidney Donation: Decomposition My probability instructor’s favorite example... Kidney donations at a hospital follow a Poisson Process of rate λ per day. Each kidney either comes from blood type A or blood type B , with probabilities p and ( 1 − p ) respectively. " All ÷H¥H÷ : xp " 19 / 22

Kidney Donation: Decomposition If I have blood type B, how long do I need to wait before receiving a compatible kidney? - p ) Ali process B rate Type Poisson : TN Expo ( XG - p ) EGF F- B until first ) time . Say I just received a type A kidney. The patient receiving a type A kidney after me is expected to live 50 more days without a kidney donation. What is the probability they survive? kidney A until next T time = . Expo ( Xp ) T - - APX 50 ) - Xpxdx - e PETE 501=150 ape I = 20 / 22

Kidney Donation: Decomposition Now imagine kidneys are types A, B, O with probabilities p , q , ( 1 − p − q ) , respectively. If I have type B blood, I can receive both B and O. How many compatible kidneys do I expect to see over the next 3 days? XXXI # . - q ) - p Xu { go.tt#Bokidn..eys.?days-PoiC3aq)EE3If7p , - op ) poi ( 37ft - p ~ 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