Poisson Distribution: Review Poisson Over Time Let B 1 ∼ Poisson( λ ) be the number of bikes that Values: are stolen on campus in one hour. (Go bears?) The Poisson Arrival Process Parameter(s) : What is the distribution of B 2 . 5 , the number of CS 70, Summer 2019 P [ X = i ] = bikes that are stolen on campus in two hours? Bonus Lecture, 8/14/19 E [ X ] = Var[ X ] = Rate over time 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? Let T 2 ∼ Poisson( λ 2 ) be the number of particles Each particle behaves independently of others. detected by Machine 2 over 4 hours. Each detected particle is an α -particle with The machines run independently . probability p , and a β -particle otherwise. What is the distribution of T 1 + T 2 ? Let T α be the number of α -particles detected by a machine over one hour. What is its distribution? 4 / 22 5 / 22 6 / 22
Decomposing Poissons Independence? Decomposing Poissons Remix 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 others, and is α with probability p , β with probability q , and γ otherwise. T α ∼ T β ∼ T γ ∼ Punt: T α , T β , T γ are mutually independent . How about T β , the number of β -particles? Sanity Check: T α + T β + T γ ∼ 7 / 22 8 / 22 9 / 22 Exponential Distribution: Review Break Poisson Arrival Process Properties We’ll now work with a specific setup: Values: ◮ There are independent “arrivals” over time. ◮ The time between consecutive arrivals is Parameter(s) : Expo( λ ) . We call λ the rate . If you could rename the Poisson RV (or any RV P [ X = i ] = Times between arrivals also independent . for that matter), what would you call it? ◮ For a time period of length t , the number of arrivals in that period is Poisson( λ t ) . E [ X ] = ◮ Disjoint time intervals have independent Var[ X ] = numbers of arrivals. 10 / 22 11 / 22 12 / 22
Poisson Arrival Process: A Visual Transmitters I Transmitters I 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? Given that I’ve seen 0 messages at time t , what is the expected time until I see the first? 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 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? We receive messages from both A and B . We receive messages from both A and B . What is the expected amount of time until the What is the expected amount of time until the first message from either transmitter? first message from either transmitter? 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. Kidney donations at a hospital follow a Poisson Process of rate λ per day. Each kidney either If I have type B blood, I can receive both B and O. 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 expected to live 50 more days without a kidney donation. What is the probability they survive? 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
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