Question on exponential distribution?

In summary, the conversation discusses a problem involving an exponential distribution, specifically calculating the probability of X being less than a certain value. Initially, the person thought the solution would be P(X <= 2) using the Law of Memoryless, but it turns out the correct solution is P(X <= 4.5) - P(X <= 2.5). This is because the Poisson distribution, which was initially mentioned, is discrete and does not apply in this situation. The correct equation for the exponential distribution is f(x) = λe-λx for x ≥ 0. The solution is found using the cumulative distribution function, F(x), and subtracting the probability of X being less than a certain value that is not wanted
  • #1
theBEAST
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Homework Statement


BR73C0s.png


Homework Equations


f(x) = eλx/x!

The Attempt at a Solution


Initially I thought I could solve this problem using the Law of Memoryless. That, the solution would just be P(X <= 2). However, I was wrong. Turns out the solution is P(X <= 4.5) - P(X<= 2.5). Does anyone know why?
 
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  • #2
First of all the equation you listed is actually a poisson distribution. As the Poisson distribution is discrete, it really wouldn't make sense in this situation.

http://en.wikipedia.org/wiki/Exponential_distribution

The an exponential distribution has pdf of the form:

f(x) = λe-λx for x ≥ 0.

Anyway, what you said isn't quite right.

Turns out the solution is P(X = 4.5) - P(X=2.5)

Since this is a continuous distribution, P(X = 4.5) = 0. In fact, P(X = c) = 0 for any c.

The solution would actually be P(X ≤ 4.5) - P(X ≤ 2.5) = F(4.5) - F(2.5).

Where F(x) is the cumulative distribution function.

The reason for this is that you want X to be less that 4.5, but all the stuff below 2.5 is not wanted , so you subtract off the probability that X is less that 2.5.
 
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FAQ: Question on exponential distribution?

What is the exponential distribution?

The exponential distribution is a probability distribution that describes the time between events in a Poisson process, where events occur continuously and independently at a constant average rate. It is often used to model the time between occurrences of events such as radioactive decay, traffic accidents, or customer arrivals.

What is the formula for the exponential distribution?

The formula for the exponential distribution is f(x) = λe^(-λx), where λ is the rate parameter and x is the random variable representing the time between events. This formula can also be written in terms of the mean (μ) as f(x) = (1/μ)e^(-x/μ).

How is the exponential distribution different from other distributions?

The exponential distribution differs from other distributions in that it is a continuous distribution, meaning that the random variable can take on any value within a defined range. It also has a single parameter, the rate parameter λ, which determines the shape of the distribution. Additionally, the exponential distribution is skewed to the right, with a long tail on the positive side.

What is the expected value of the exponential distribution?

The expected value, or mean, of the exponential distribution is 1/λ. This means that on average, the time between events in a Poisson process will be 1/λ units of time. For example, if λ = 2, then the average time between events would be 0.5 units of time.

How is the exponential distribution used in real-world applications?

The exponential distribution is commonly used in many fields, including physics, biology, economics, and engineering. It can be used to model the lifetimes of products, the time between equipment failures, or the time between customer arrivals in a queue. It is also used in survival analysis to estimate the probability of an event occurring at a certain time.

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