Probability

Consider a Poisson process with rate lambda = 2 and let T be the time of the first arrival.

1. Find the conditional PDF of T given that the second arrival came before time t = 1. Enter an expression in terms of lambda and t.

2. Find the conditional PDF of T given that the third arrival comes exactly at time t = 1.

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  1. Does anyone have the answer please?

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  2. The answer is you have to work harder on your studies, better yourself and not cheat. Btw does anyone have the answer please

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  3. 1) i am not sure

    lambda*e^(-lambda)*(e^(-lambda*(t-1))-1)/(1-(1+lambda)*e^(-lambda))

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  4. 1) (lambda*exp(-lambda*t) - lambda*exp(-lambda)) / (1-(lambda+1)*exp(-lambda))

    2) 2*(1-t)

    Both can be obtained using Bayes' rule P(A|B) = P(A)P(B|A) / P(B). I cannot say for sure they are right, but as a first test their integral from 0 to 1 (domain for t) is 1 so.. let's cross the fingers!

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  5. I think the second part is 3t^2 - 6t + 3.

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  6. 1. 2*t*(1-t)
    Given P(A⋂B) = λtexp(-λt)*λ(1-t)exp(-λ(1-t))
    P(B)=λ²exp(-λ)/2!
    2. 3*t*(1-t)²
    Given P(A⋂B) = λtexp(-λt)*λ²(1-t)^2exp(-λ(1-t))
    P(B)=λ^3exp(-λ)/3!

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  7. Is the first part supposed to be a constant? Is that what I should be looking for?

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  8. I think my calculation for the second part was wrong. If the third arrival comes at t=1, then the first two arrivals are uniformly distributed on [0, 1], so we should have E[T] = 1/3. Recalculating based on the order statistics gives 2 - 2t, which matches RocknRoll's answer above and gives the right expected value.

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