Stats/Probability: Joint Exponential Distribution

Homework Statement

Suppose that X=time to failure for a component has an exponential distribution with lambda =.25. Suppose that 9 of the components are selected and their failure times noted. Compute the probability that 3 of the components fail between times 1 and 2, and 4 of the components fail between times 2 and 3. Assume that the failure times are independent.

Homework Equations

Exponential Distribution: f(x;λ)=λe-xλ

The Attempt at a Solution

F(x,y)=∫∫0.25(e^(-x/4)*e^(-y/4))dxdy

I solved these over the intervals 2 to 1 and 3 to 2, and came up with F(1<x<2,2<y<3)=0.0925 as a solution for the probability that a component will fail. However, I'm not sure how to apply this to the 3/9 and 4/9 attempts. Am I even on the right track here?

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Ray Vickson
Homework Helper
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Homework Statement

Suppose that X=time to failure for a component has an exponential distribution with lambda =.25. Suppose that 9 of the components are selected and their failure times noted. Compute the probability that 3 of the components fail between times 1 and 2, and 4 of the components fail between times 2 and 3. Assume that the failure times are independent.

Homework Equations

Exponential Distribution: f(x;λ)=λe-xλ

The Attempt at a Solution

F(x,y)=∫∫0.25(e^(-x/4)*e^(-y/4))dxdy

I solved these over the intervals 2 to 1 and 3 to 2, and came up with F(1<x<2,2<y<3)=0.0925 as a solution for the probability that a component will fail. However, I'm not sure how to apply this to the 3/9 and 4/9 attempts. Am I even on the right track here?
The question's wording is a bit ambiguous. I can't figure out if you are being asked for (i) the probability that 3 fail between times 1 and 2, and (ii)[separately] the probability that 4 fail between times 2 and 3, or whether you are being asked for the joint probability that 3 fail between 1 and 2 and 4 fail between 2 and 3 (all considered as a single event).

Anyway, it is for such questions that the Poisson distribution was invented. Google Poisson process and Poisson distribution. The "memoryless" property of the exponential distribution is also very important in such cases.

RGV

It's meant to be as a joint probability. I'm a little confused by your response though, I thought Poisson was mostly used for when you're only given a mean/average success rate and need to solve using it? I'm assuming here you mean to find this rate and then apply it?

In the case of an exponential, μ=1/λ so the average time to fail should be 4, correct? So then I would plug that into a Poisson formula to get a solution?

Ray Vickson
Homework Helper
Dearly Missed
It's meant to be as a joint probability. I'm a little confused by your response though, I thought Poisson was mostly used for when you're only given a mean/average success rate and need to solve using it? I'm assuming here you mean to find this rate and then apply it?

In the case of an exponential, μ=1/λ so the average time to fail should be 4, correct? So then I would plug that into a Poisson formula to get a solution?
OK, never mind about the Poisson stuff.

Start by considering a simpler case than the one you are given. For example, suppose you ask for the distribution of the number that fail at times between 1 and 2. How would you get that? Now apply the same reasoning to your original problem.

RGV