How Is E(S5) Calculated in a Poisson Process?

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In summary, the problem involves a Poisson process with rate λ and the task is to find the expected value of the fifth occurrence, given that the first occurrence is known to be 5. The solution is not as straightforward as simply dividing n by λ, as there are other factors to consider.
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CTK
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Summary: inter-arrival times problem

Let {X(t) : t ≥ 0} be a Poisson process with rate λ.

a-) Let Si denote the time of the ith occurrence, i = 1, 2, . . . . Suppose it is known that X(1) = 5. Find ## E(S_{5}) ##.

My attempt: Is the answer simply ## n / \lambda = 5 / \lambda ##? Or is there more to it that I am missing?

Thanks for the help
 
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  • #2
CTK said:
Summary: inter-arrival times problem

Let {X(t) : t ≥ 0} be a Poisson process with rate λ.

a-) Let Si denote the time of the ith occurrence, i = 1, 2, . . . . Suppose it is known that X(1) = 5. Find ## E(S_{5}) ##.

My attempt: Is the answer simply ## n / \lambda = 5 / \lambda ##?
Why would that be the answer?
 
  • #3
PeroK said:
Why would that be the answer?
Hi PeroK. I guess I have just figured it out, so no worries. Thanks for your reply.
 

FAQ: How Is E(S5) Calculated in a Poisson Process?

1. What is the Inter-arrival times problem?

The Inter-arrival times problem is a statistical problem that involves analyzing the time intervals between consecutive occurrences of a particular event. It is commonly used in fields such as queueing theory, reliability analysis, and time series analysis.

2. What is the significance of the Inter-arrival times problem?

The Inter-arrival times problem is important because it can provide insights into the behavior and patterns of events over time. By analyzing the inter-arrival times, we can make predictions and improve processes in various industries, such as transportation, healthcare, and telecommunications.

3. How is the Inter-arrival times problem solved?

The Inter-arrival times problem can be solved using various statistical methods, such as probability distributions, regression analysis, and time series models. The choice of method depends on the nature of the data and the specific objectives of the analysis.

4. What are some common applications of the Inter-arrival times problem?

The Inter-arrival times problem has many applications, including predicting customer arrivals in a queue, estimating the time between equipment failures, and forecasting the arrival times of buses or trains. It is also used in quality control to identify patterns in the occurrence of defects.

5. What are some challenges in analyzing the Inter-arrival times problem?

One of the main challenges in analyzing the Inter-arrival times problem is dealing with incomplete or biased data. Another challenge is selecting the appropriate statistical method and interpreting the results accurately. Additionally, the complexity of the problem may increase with the number of events and the variability of the inter-arrival times.

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