Calculating Probability in Poisson Process Problem | Z(t-c)=m, Z(t)=k

Thread starter alehand12
Start date Jun 21, 2011
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Poisson Poisson process Process

In summary, a Poisson process is a mathematical model used to describe the behavior of random events occurring over time. Its key characteristics include independent events, a constant rate, and probability proportional to interval length. It differs from other stochastic processes and can be used to solve problems related to event probability and waiting time. Real-world applications include queuing theory, telecommunications, finance, and actuarial science.

Jun 21, 2011

alehand12

Given a poisson process Z(t) with a given rate lamda, k and m nonnegative integers and t and c real and positive numbers, calculate the probability:
P(Z(t-c)=m | Z(t)=k)

thanks

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Jun 22, 2011

bpet

Hint: conditional on the number of jumps in a Poisson process, the jump times are uniformly distributed over the time interval.

Related to Calculating Probability in Poisson Process Problem | Z(t-c)=m, Z(t)=k

What is a Poisson process?

A Poisson process is a mathematical model used to describe the behavior of random events occurring over time. It assumes that the events occur independently of each other and at a constant rate.

What are the key characteristics of a Poisson process?

The key characteristics of a Poisson process are that the events occur independently of each other, the rate at which events occur is constant, and the probability of an event occurring in a given time interval is proportional to the length of the interval.

How is a Poisson process different from other stochastic processes?

A Poisson process differs from other stochastic processes in that it models the occurrence of discrete events over time, while other processes such as Brownian motion or Markov processes model continuous changes over time.

What types of problems can be solved using a Poisson process?

A Poisson process can be used to solve problems related to the probability of a certain number of events occurring in a given time period, the probability of a specific number of events occurring in a certain order, and the expected waiting time between events.

What are some real-world applications of Poisson processes?

Poisson processes are commonly used in fields such as queuing theory, telecommunications, finance, and actuarial science to model and analyze random events and their impact on systems or processes. For example, they can be used to model the arrival of customers at a bank or the occurrence of accidents on a highway.