What is the Probability of Two Major Cracks in a 10 Mile Stretch of Highway?

In summary, the conversation discusses the use of an exponential distribution with a mean of 5 miles to calculate the probability of two major cracks occurring in a 10 mile stretch of highway. The solution involves using the Poisson process with a lambda value of 2, resulting in a probability of 0.2706.
  • #1
Andrew123
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Homework Statement



The distance between major cracks in the highway follows an exponential distribution with a mean of 5 miles. What is the probability that there are two major cracks in a 10 mile stretch of the highway?


Homework Equations



exponential dist: f(x) = Le(-Lx) where L = lambda (sorry i don't know latex)

E(X) = 1/L V(X) = 1/(L)^2

The Attempt at a Solution



well I'm abit stumped because the probability density function for the exponential dist tells us the time to an event etc.. not the number of events. So i really don't have much of an attempt :(
 
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  • #2
got it.. just use the poisson process.. with lambda = 2 etc.. answer is 0.2706
 

What is the Exponential Distribution?

The Exponential Distribution is a probability distribution that models the time between events in a Poisson process. It is commonly used to model the time between radioactive decay events, the time between arrivals of customers at a store, or the time between earthquakes.

What are the characteristics of the Exponential Distribution?

The Exponential Distribution has a single parameter, λ (lambda), which measures the rate at which events occur. It is a continuous distribution, meaning that the possible values it can take on are infinite and uncountable. It is also a skewed distribution, with a long right tail.

How is the Exponential Distribution different from other probability distributions?

The Exponential Distribution is unique in that it is the only continuous probability distribution that has a constant failure rate (also known as the hazard rate). This means that the probability of an event occurring in a given time period remains the same, regardless of how much time has already passed.

What are some real-life applications of the Exponential Distribution?

The Exponential Distribution is commonly used in reliability engineering, to model the time between failures of a system. It is also used in queuing theory, to model the time between arrivals of customers at a store or service center. Additionally, it is used in finance, to model the time between stock price changes and in epidemiology, to model the time between disease outbreaks.

How do you calculate probabilities and expected values for the Exponential Distribution?

To calculate probabilities and expected values for the Exponential Distribution, you can use the following formulas:

  • The probability density function (PDF) is f(x) = λe^(-λx), where x is the time between events and λ is the rate parameter.
  • The cumulative distribution function (CDF) is F(x) = 1 - e^(-λx).
  • The expected value (mean) is E(x) = 1/λ.

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