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.

1. 👍
2. 👎
3. 👁
4. ℹ️
5. 🚩

1. 👍
2. 👎
3. ℹ️
4. 🚩
2. The answer is you have to work harder on your studies, better yourself and not cheat. Btw does anyone have the answer please

1. 👍
2. 👎
3. ℹ️
4. 🚩
3. 1) i am not sure

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

1. 👍
2. 👎
3. ℹ️
4. 🚩
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!

1. 👍
2. 👎
3. ℹ️
4. 🚩
5. I think the second part is 3t^2 - 6t + 3.

1. 👍
2. 👎
3. ℹ️
4. 🚩
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!

1. 👍
2. 👎
3. ℹ️
4. 🚩
7. Is the first part supposed to be a constant? Is that what I should be looking for?

1. 👍
2. 👎
3. ℹ️
4. 🚩
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.

1. 👍
2. 👎
3. ℹ️
4. 🚩

Similar Questions

1. Mathematics

Regular (not junk) emails arrive at your inbox according to a Poisson process with rate r; and junk emails arrive at your inbox according to an independent Poisson process with rate j. Assume both processes have been going on

2. Probability

All ships travel at the same speed through a wide canal. Each ship takes days to traverse the length of the canal. Eastbound ships (i.e., ships traveling east) arrive as a Poisson process with an arrival rate of ships per day.

3. Probability

Consider a Poisson process with rate λ. Let N be the number of arrivals in (0,t] and M be the number of arrivals in (0,t+s], where t>0,s≥0. In each part below, your answers will be algebraic expressions in terms of λ,t,s,m

4. Probability

In parts 1, 3, 4, and 5 below, your answers will be algebraic expressions. Enter 'lambda' for and 'mu' for . Follow standard notation. 1. Shuttles bound for Boston depart from New York every hour on the hour (e.g., at exactly one

1. probability

1. Busy people arrive at the park according to a Poisson process with rate λ1=3/hour and stay in the park for exactly 1/6 of an hour. Relaxed people arrive at the park according to a Poisson process with rate λ2=2/hour and stay

2. probability

Based on your understanding of the Poisson process, determine the numerical values of a and b in the following expression. ∫∞tλ6τ5e−λτ5!dτ=∑k=ab(λt)ke−λtk!. a= ? b= ?

3. probability

As in an earlier exercise, busy people leave the park according to a Poisson process with rate λ1=3/hour. Relaxed people leave the park according to an independent Poisson process with rate λ2=2/hour. Each person, upon leaving

4. probability

Consider a Poisson arrival process with rate λ per hour. To simplify notation, we let a=P(0,1), b=P(1,1), and c=P(2,1), where P(k,1) is the probability of exactly k arrivals over an hour-long time interval. What is the

1. Statistics & Probability

Consider a Poisson process with rate λ=4, and let N(t) be the number of arrivals during the time interval [0,t]. Suppose that you have recorded this process in a movie and that you play this movie at twice the speed. The process

2. Probability

Events related to the Poisson process can be often described in two equivalent ways: in terms of numbers of arrivals during certain intervals or in terms of arrival times. The first description involves discrete random variables,

3. Statistics & Probability

Consider a Poisson process with rate λ=1. Consider three times that satisfy 0

4. Statistics & Probability

Let Yk be the time of the k th arrival in a Poisson process with parameter λ=1 . In particular, E[Yk] = k . 1. Is it true that P(Yk≥k) = 1/2 for any finite k ? 2. Is it true that limk→∞P(Yk≥k) = 1/2 ?