(a) Radioactive particles are detected by a counter according to a Poisson process with rate parameter λ = 0.5 particles per second. What is the probability that two particles are detected in any given one second interval ? (b) Page visits to a particular website occur according to a Poisson process with rate parameter λ = 20 per minute. What is the expected number of visits to the website in any given one hour period ? (c) 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 ) > 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)“x!x = 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 = #10X2 = x2) = P(X1 = #1)P(X2 = x2)%3DwithX1 ~ 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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(a) Radioactive particles are detected by a counter according to a Poisson process with rate parameter λ = 0.5 particles per second. What is the probability that two particles are detected in any given one second interval ?

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

(a) Radioactive particles are detected by a counter according to a Poisson process with rate parameter λ = 0.5 particles per second. What is the probability that two particles are detected in any given one second interval ?
(b) Page visits to a particular website occur according to a Poisson process with rate parameter λ = 20 per minute. What is the expected number of visits to the website in any given one hour period ?
(c) 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 ) > 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)“
x!
x = 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 = #10X2 = x2) = P(X1 = #1)P(X2 = x2)
%3D
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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