🔀Stochastic Processes Unit 4 – Poisson processes

Key Concepts and Definitions

• Poisson process models the occurrence of events over time or space when events happen independently at a constant average rate
• Characterized by the Poisson distribution, which gives the probability of a specific number of events occurring in a fixed interval
• Defined by a single parameter $\lambda$, the average rate of events per unit time or space
• Memoryless property implies that the probability of an event occurring does not depend on the time since the last event
• Poisson process is a continuous-time Markov chain with an infinite number of states
• Poisson distribution approximates the binomial distribution when the number of trials is large and the probability of success is small
• Poisson process is a special case of a renewal process where interarrival times are exponentially distributed

Properties of Poisson Processes

• Events occur one at a time and independently of each other
• The average rate of events remains constant over time or space
• The number of events in disjoint intervals is independent
• The probability of an event occurring in a small interval is proportional to the length of the interval
  • Mathematically, $P(N(t+h) - N(t) = 1) = \lambda h + o(h)$, where $o(h)$ is a function that grows slower than $h$
• The number of events in an interval of length $t$ follows a Poisson distribution with parameter $\lambda t$
• The waiting time between consecutive events (interarrival time) is exponentially distributed with parameter $\lambda$
• Splitting a Poisson process results in independent Poisson processes (splitting property)

Probability Distribution and Calculations

• The probability mass function (PMF) of a Poisson distribution is given by $P(X = k) = \frac{e^{-\lambda t}(\lambda t)^k}{k!}$, where $X$ is the number of events in an interval of length $t$
• The mean and variance of a Poisson distribution are both equal to $\lambda t$
• The cumulative distribution function (CDF) of a Poisson distribution is given by $P(X \leq k) = e^{-\lambda t} \sum_{i=0}^k \frac{(\lambda t)^i}{i!}$
• The moment-generating function (MGF) of a Poisson distribution is $M_X(s) = e^{\lambda t(e^s - 1)}$
• The sum of independent Poisson random variables is also a Poisson random variable with a rate equal to the sum of the individual rates
• The probability of no events occurring in an interval of length $t$ is $P(X = 0) = e^{-\lambda t}$
• The probability of at least one event occurring in an interval of length $t$ is $P(X \geq 1) = 1 - e^{-\lambda t}$

Interarrival Times and Exponential Distribution

• Interarrival times in a Poisson process are independently and identically distributed (i.i.d.) exponential random variables with parameter $\lambda$
• The probability density function (PDF) of an exponential distribution is given by $f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$
• The CDF of an exponential distribution is $F(x) = 1 - e^{-\lambda x}$ for $x \geq 0$
• The mean and standard deviation of an exponential distribution are both equal to $\frac{1}{\lambda}$
• The memoryless property of the exponential distribution states that $P(X > s + t | X > s) = P(X > t)$ for all $s, t \geq 0$
  • This implies that the waiting time until the next event does not depend on how long one has already waited
• The sum of independent exponential random variables follows a gamma distribution
• The minimum of independent exponential random variables is also an exponential random variable with a rate equal to the sum of the individual rates

Compound Poisson Processes

• A compound Poisson process is a generalization of the standard Poisson process where each event is associated with a random variable (mark)
• The marks are i.i.d. random variables, independent of the Poisson process itself
• The total mark accumulation up to time $t$ is given by $S(t) = \sum_{i=1}^{N(t)} Y_i$, where $N(t)$ is the number of events up to time $t$ and $Y_i$ are the marks
• The mean of the compound Poisson process at time $t$ is $E[S(t)] = \lambda t E[Y]$, where $E[Y]$ is the mean of the mark distribution
• The variance of the compound Poisson process at time $t$ is $Var[S(t)] = \lambda t E[Y^2]$, where $E[Y^2]$ is the second moment of the mark distribution
• Examples of compound Poisson processes include the total claim amount in an insurance portfolio and the total number of customers served in a queue with random service times

Applications and Real-World Examples

• Modeling the number of customers arriving at a store or calls arriving at a call center per hour (retail, customer service)
• Analyzing the number of defects or failures in a manufacturing process or equipment (quality control, reliability engineering)
• Studying the number of accidents or claims in an insurance portfolio (actuarial science)
• Investigating the number of mutations in a DNA sequence over time (genetics, bioinformatics)
• Modeling the number of particles emitted by a radioactive source (nuclear physics)
• Analyzing the number of requests or packets arriving at a server or network (computer science, telecommunications)
• Predicting the number of rare events, such as earthquakes or volcanic eruptions, in a given time period (geology, seismology)

Problem-Solving Techniques

• Identify the rate parameter $\lambda$ based on the given information or data
• Determine the appropriate probability distribution (Poisson, exponential, or compound Poisson) for the problem
• Use the PMF or CDF to calculate probabilities for specific events or intervals
• Apply the properties of Poisson processes, such as independence and memoryless property, to simplify calculations
• Utilize the relationship between Poisson and exponential distributions for problems involving interarrival times
• Break down compound Poisson process problems into separate components (Poisson process and mark distribution) and combine the results
• Apply the Poisson approximation to the binomial distribution when appropriate (large number of trials, small probability of success)
• Use conditioning and the law of total probability to solve problems involving multiple Poisson processes or events

Related Stochastic Processes

• Renewal processes generalize Poisson processes by allowing interarrival times to follow any distribution, not just exponential
• Birth-death processes model populations where individuals are born and die according to Poisson processes
• Queueing theory heavily relies on Poisson processes to model the arrival of customers or requests in various queueing systems (M/M/1, M/M/c, etc.)
• Markov chains can be used to model Poisson processes by discretizing time and considering the number of events in each time step
• Brownian motion can be obtained as a limit of a scaled Poisson process (or a compound Poisson process with normally distributed marks) as the rate goes to infinity
• Hawkes processes are self-exciting point processes where the occurrence of an event increases the likelihood of future events, generalizing the constant rate assumption of Poisson processes
• Poisson point processes extend the concept of Poisson processes to higher dimensions, modeling the spatial distribution of events in a plane or space