Poisson Process and Stress Fractures in Railway Lines

In summary, we are given a Poisson process with a rate of 2 stress fractures per month on railway lines. Part a asks for the probability that the 4th stress fracture occurs 3 months after checking the lines. Using the Poisson probability formula, we get a result of (54e^-6)/3!. In part b, we are asked how long it is expected to take for the 4th stress fracture to occur on new railway lines. Again using the Poisson probability formula, we get a result of (512e^-8)/4!. There is no confidence level given for part b.
  • #1
Jenny123
2
0

Homework Statement


Suppose that stress fractures appear in railway lines according to a Poisson process at a rate of 2 per month.
a)Find the probability that the 4th stress fracture on the railway line occurred 3 months after the process of checking the new railway lines.

b)Suppose new railway lines have just been laid, how long (in months) is it expected to take for the 4th stress factor to occur?


Homework Equations





The Attempt at a Solution


a) P(N(3) =3) = (54 exp{-6} )/3!

b) P(N(4) =4) = (512 exp{-8})/4!
 
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  • #2
What is your formula for the probability function? I don't see how you get 54 in part a.
You appear to use the same probability estimation to determine the number of months in b. Is there a confidence level for b?
 

Related to Poisson Process and Stress Fractures in Railway Lines

What is a Poisson Process?

A Poisson Process is a mathematical model used to describe the occurrence of random events over a continuous period of time. It assumes that events occur independently of each other and at a constant rate.

How do Poisson Processes relate to stress fractures in railway lines?

In the context of railway lines, a Poisson Process can be used to model the occurrence of stress fractures due to the random and independent nature of their formation. This information can then be used to predict the likelihood and timing of future stress fractures, allowing for more effective maintenance and prevention strategies.

What factors can contribute to the formation of stress fractures in railway lines?

Several factors can contribute to the formation of stress fractures in railway lines, including heavy train loads, extreme temperature changes, poor track maintenance, and improper track design. These factors can lead to excessive stress and strain on the railway line, causing it to weaken and eventually fracture.

How can the risk of stress fractures in railway lines be reduced?

The risk of stress fractures in railway lines can be reduced by regularly inspecting and maintaining the tracks, using materials that can withstand heavy loads and temperature changes, and implementing proper track design and construction techniques. Additionally, utilizing data from a Poisson Process model can help identify high-risk areas and prioritize maintenance efforts.

What role do scientists play in studying and addressing stress fractures in railway lines?

Scientists play a crucial role in studying and addressing stress fractures in railway lines by conducting research to better understand the causes and mechanisms of fracture formation, developing and testing new materials and construction techniques, and using mathematical models like Poisson Processes to predict and prevent future stress fractures. Their work is crucial in ensuring the safety and efficiency of railway systems.

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