Poisson Processes (9) Flashcards
(18 cards)
What is a Poisson process?
A Poisson process is a continuous time Markov process that counts the number of events in (0, t] where events occur independently at a constant rate λ.
What is the state space of a Poisson process?
The state space of a Poisson process is infinite and consists of the natural numbers together with {0}: S = {0, 1, 2, 3, …}.
How is a Poisson process defined?
A Poisson process is defined by the property that the number of events in (t, t+h] follows P(1 event in (t, t+h]) = λh + o(h), and the number of events in (t, t+h] is independent of the history before time t.
What is the holding time in a Poisson process?
The holding time in a Poisson process is exponentially distributed with parameter λ, meaning the time between events follows an exponential distribution.
What is the rate of leaving state i in a Poisson process?
The rate of leaving state i in a Poisson process is λ, as each event occurs at rate λ.
What is the generator matrix for a Poisson process?
The generator matrix for a Poisson process is Q = [ -λ λ 0 0 …; 0 -λ λ 0 …; 0 0 -λ λ …; … ] This is because the rate of leaving state i to only state i+1 (poisson process can only go to one state at a time) is lambda. The negative lambda means we are leaving that state.
What is the distribution of the number of events by time t?
The distribution of the number of events by time t follows a Poisson distribution with parameter λt, i.e., N(t) ∼ Poisson(λt).
What is the probability that k events occur in (0,t] in a Poisson process?
The probability that k events occur in (0,t] is given by p_0k(t) = P(N(t) = k) = (λt)^k * exp(-λt) / k!
What is the distribution of the number of events in (s,s+t]?
The number of events in (s, s+t] follows a Poisson(λt) distribution, and the distribution is stationary over time.
What is the time to the first event in a Poisson process?
The time to the first event in a Poisson process is exponentially distributed with parameter λ, with P(T1 > t) = exp(-λt).
What is the distribution of the times between successive events in a Poisson process?
The times between successive events in a Poisson process are independent and exponentially distributed with rate λ.
What is the time to the rth event in a Poisson process?
The time to the rth event in a Poisson process follows a Gamma(r, λ) distribution, as it is the sum of r independent exponential random variables with rate λ.
What is the time from an arbitrary time to the next event in a Poisson process?
The time from an arbitrary time to the next event in a Poisson process is exponentially distributed with rate λ, due to the memoryless property.
What is the conditional distribution of times of events given the number of events?
Given exactly k events occur in (0,t], the arrival times of those events are uniformly distributed over (0,t). This follows because the events are scattered randomly over the interval.
What happens when we superpose two Poisson processes?
When two Poisson processes with rates λ1 and λ2 are independent, their superposition is also a Poisson process with rate λ1 + λ2.
What happens when we thin a Poisson process?
When thinning a Poisson process, each event is kept with probability p and deleted with probability 1-p. The resulting process is a Poisson process with rate pλ.
What is the generator matrix for the charity worker Poisson process?
The generator matrix for the charity worker Poisson process is Q = [ -2 2; 1 -1 ] due to the rate of arrivals and the independent thinning of the process.
How do you find the long-run proportion of time that the charity worker is busy?
The long-run proportion of time that the charity worker is busy can be found by solving πQ = 0, yielding π1 = 2/3.
