Probability distribution of first arrival time in Poisson Process

In summary, a Poisson process is a stochastic process that models the occurrence of events over time. The first arrival time in a Poisson process is calculated using a mathematical formula and is significant in analyzing and predicting the behavior of systems with random events. The mean and variance of a Poisson distribution affect the first arrival time, and it can be used to predict future events, although accuracy may vary due to the randomness of the process.
  • #1
phyalan
22
0
According to wiki:
http://en.wikipedia.org/wiki/Poisson_process

The probability for the waiting time to observe first arrival in a Poisson process P(T1>t)=exp(-lambda*t)
But what is the Probability Distribution P(T1=t) of the waiting time itself? How to calculate that?
 
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  • #2
Since the waiting time has a continuous distribution, the probability of T1 having a particular value = 0. I suppose you would want the density function, λe-λt.
 

1. What is a Poisson process and how does it relate to probability distribution?

A Poisson process is a stochastic process that models the occurrence of events over time. It is used to describe random events that happen independently of each other and at a constant rate. The probability distribution of first arrival time in a Poisson process is a mathematical function that describes the likelihood of the first event occurring at a specific time.

2. How is the first arrival time calculated in a Poisson process?

The first arrival time in a Poisson process is calculated using the Poisson distribution formula, which takes into account the rate of occurrences and the time interval. The formula is P(X = x) = (λt)^x * e^(-λt) / x!, where λ is the rate and t is the time interval.

3. What is the significance of the first arrival time in a Poisson process?

The first arrival time in a Poisson process is important as it represents the time it takes for the first event to occur in a series of random events. It is used to analyze and predict the behavior of systems that involve random events, such as customer arrivals in a queue or the occurrence of natural disasters.

4. How does the mean and variance of a Poisson distribution affect the first arrival time?

The mean and variance of a Poisson distribution directly affect the first arrival time. The mean, represented by λ, determines the rate of occurrences and therefore affects how frequently the first event will occur. The variance, represented by σ^2 = λt, affects the spread of the probability distribution and therefore impacts the range of possible arrival times for the first event.

5. Can the first arrival time in a Poisson process be used to predict future events?

Yes, the first arrival time in a Poisson process can be used to predict future events. By analyzing the probability distribution, we can estimate the likelihood of the first event occurring at a certain time and use this information to make predictions about future events. However, it is important to note that due to the randomness of a Poisson process, these predictions may not always be accurate.

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