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.
8 answers
Does anyone have the answer please?
The answer is you have to work harder on your studies, better yourself and not cheat. Btw does anyone have the answer please
1) i am not sure
lambda*e^(-lambda)*(e^(-lambda*(t-1))-1)/(1-(1+lambda)*e^(-lambda))
lambda*e^(-lambda)*(e^(-lambda*(t-1))-1)/(1-(1+lambda)*e^(-lambda))
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!
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!
I think the second part is 3t^2 - 6t + 3.
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!
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!
Is the first part supposed to be a constant? Is that what I should be looking for?
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.
1 answer