What is the probability that the time of the first event that is observed to occur in a Poisson process with rate λ per unit time, after initiation at t = 0, occurs later than time t = t0, for fixed value t0 ? Justify your answer The Poisson process is a model for events that occur in continuous time, at a constant rate A > 0 perunit time, with events occurring independently of each other. Specifically, if X(t) is the discreterandom variable recording the number of events that are observed to occur in the interval [0, t),then we have that X(t) ~ Poisson(At), that isP(x) = P(X(t) = x) = e¬At (At)=T = 0, 1, 2, ...and zero otherwise. Also, the counts of events in disjoint time intervals are probabilisticallyindependent: for example, for intervals [0, t) and [t, t'+ s), the numbers of events in the twointervals, X1 and X2 say, have the propertyP(X1 = x1n X2 = x2) = P(X1 = x1)P(X2= x2)withX1 ~ Poisson(At)X2 - Poisson(As).With this information, answer the following questions based on the Poisson process model andits relationship with the Poisson distribution.

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What is the probability that the time of the first event that is observed to occur in a Poisson process with rate λ per unit time, after initiation at t = 0, occurs later than time t = t0, for fixed value t0 ? Justify your answer

What is the probability that the time of the first event that is observed to occur in a Poisson process with rate λ per unit time, after initiation at t = 0, occurs later than time t = t0, for fixed value t0 ? Justify your answer

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Question

The Poisson process is a model for events that occur in continuous time, at a constant rate A > 0 per
unit time, with events occurring independently of each other. Specifically, if X(t) is the discrete
random variable recording the number of events that are observed to occur in the interval [0, t),
then we have that X(t) ~ Poisson(At), that is
P(x) = P(X(t) = x) = e¬At (At)=
T = 0, 1, 2, ...
and zero otherwise. Also, the counts of events in disjoint time intervals are probabilistically
independent: for example, for intervals [0, t) and [t, t'+ s), the numbers of events in the two
intervals, X1 and X2 say, have the property
P(X1 = x1n X2 = x2) = P(X1 = x1)P(X2= x2)
with
X1 ~ Poisson(At)
X2 - Poisson(As).
With this information, answer the following questions based on the Poisson process model and
its relationship with the Poisson distribution.

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