Stochastic Processes

Stochastic Processes

SlAM's Classics in Applied Mathematics series consists of books that were previously allowed to go out of print. These books are republished by SIAM as a professional service because they continue to be important resources for mathematical scientists. Editor-in-Chieff Robert E O'Malley, Jr., University of Washington Editorial Board Richard A. Brualdi, University of Wisconsin-Madison Herbert B. Keller, California Institute of Technology Andrzej Z. Manitius, George Mason University Ingram Olkin, Stanford University Stanley Richardson, University of Edinburgh Ferdinand Verhulst, Mathematisch Instituut, University of Utrecht Classics in Applied Mathematics C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences Johan G. F. Belinfante and Bernard Kolman, A Survey of Lie Croups and Lie Algebras with Applications and Computational Methods James M. Ortega, Numerical Analysis: A Second Course Anthony V. Fiacco and Garth P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques F. H. Clarke, Optimization and Nonsmooth Analysis George F. Carrier and Carl E. Pearson, Ordinary Differential Equations Leo Breitnan, Probability R. Bellman and G. M. Wing, An Introduction to Invariant Imbedding Abraham Berman and Robert J. Plemmons, Nonnegative Matrices in the Mathematical Sciences Olvi L Mangasarian, Nonlinear Programming *Carl Friedrich Gauss, Theory of the Combination of Observations Least Subject to Errors: Part One, Part Two, Supplement. Translated by G. W. Stewart Richard Bellman, Introduction to Matrix Analysis U. M. Ascher, R. M. M. Mattheij, and R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations K. E Brenan, S. L Campbell, and L R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations Charles L Lawson and Richard J. Hanson, Solving Least Squares Problems J. E. Dennis, Jr. and Robert B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations Richard E. Barlow and Frank Proschan, Mathematical Theory of Reliability Cornelius Lanczos, Linear Differential Operators Richard Bellman, Introduction to Matrix Analysis, Second Edition *First time in print.

Classics in Applied Mathematics (continued) Beresford N. Parlett, The Symmetric Eigenvalue Problem Richard Haberman, Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow Peter W. M. John, Statistical Design and Analysis of Experiments Tamer Ba$ar and Geert Jan Olsder, Dynamic Noncooperative Game Theory, Second Edition Emanuel Parzen, Stochastic Processes

This page intentionally left blank

Stochastic Processes

Emanuel Parzen Texas A&M University College Station, Texas

Society for Industrial and Applied Mathematics Philadelphia

Copyright ©1999 by the Society for Industrial and Applied Mathematics. This SI AM edition is an unabridged, corrected republication of the work first published by Holden-Day, Oakland, CA, 1962.

10987654321 All rights reserved. Printed in the United States of America. No pan of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688. Library of Congress Cataloging-in-Publication Data Parzen. Emanuel, 1929Stochastic processes / Emanuel Parzen. p. cm. • • (Classics in applied mathematics ; 24) An unabridged, corrected republication of the work first published by Holden-Day, Oakland, Calif., 1962. Includes bibliographical references and index. ISBN 0-89871441-9 (pbk.) 1. Stochastic processes. I. Title II. Series QA274.P37 1999 519.2--dc21 99-27282

is a registered trademark.

TO CAROL

This page intentionally left blank

Table of contents PREFACE TO THE CLASSICS EDITION PREFACE

xiii

xvii

ROLE OF THE THEORY OF STOCHASTIC PROCESSES

Statistical physics 1 Stochastic models for population growth Communication and control 2 Management science 4 Time series analysis 6 1 I

RANDOM VARIABLES AND STOCHASTIC PROCESSES

1-1 1-2 1-3 1-4 n L

2

Random variables and probability laws 8 Describing the probability law of a stochastic process The Wiener process and the Poisson process 26 Two-valued processes 35

2%

CONDITIONAL PROBABILITY AND CONDITIONAL EXPECTATION

2-1 Conditioning by a discrete random variable 4% 2-2 Conditioning by a continuous random variable 51 2-3 Properties of conditional expectation 62

3

N O R M A L PROCESSES AND COVARIANCE STATIONARY PROCESSES

3-1 The mean value function and covariance kernel of a stochastic process 66 ix

X

TABLE OP CONTENTS

3-2 3-3 3-4 3-5 3-6 i 4

Stationary and evolutionary processes 69 Integration and differentiation of stochastic processes 78 Normal processes 88 Normal processes as limits of stochastic processes 97 Harmonic analysis of stochastic processes 103

COUNTING PROCESSES AND POISSON PROCESSES

4-1 Axiomatic derivations of the Poisson process 118 4-2 Non-homogeneous, generalized, and compound Poisson processes 124 4-3 Inter-arrival times and waiting times 182 4-4 The uniform distribution of waiting times of a Poisson process 189 4-x( •) of X is defined, for any real number t, by The characteristic function < p x ( ' ) of -X" is defined, for any real number u, by tThe first integral in Eq. 1.8 is called a Stieltjes integral; for a definition of this integral, see Mod Prob, p. 233.

12

RANDOM VARIABLES AND STOCHASTIC PROCESSES

Ch 1

A random variable may not possess a finite mean, variance, or moment-generating function. However, it always possesses a characteristic function. Indeed, there is a one-to-one correspondence between distribution functions and characteristic functions. Consequently, to specify the probability law of a random variable, it suffices to state its characteristic function. Various "inversion" formulas, giving distribution functions, probability density functions, and probability mass functions explicitly in terms of characteristic functions, may be stated (see Mod Prob, Chapter 9). Here we state without proof two inversion formulas: (i) for any random variable X

(ii) if the characteristic function 0} defined by X(t) = A cos ut + B sin ««, (2.6) where the frequency o> is a known positive constant, and A and B are independent normally distributed random variables with means 0 and variances or2. Sufficient information is then at hand to find the probability of any assertion concerning the stochastic process; for example, let us find, for any constant c, Now, letting L = 2ir/w,

Therefore,

where FFT and IFFT denote, respectively, fast Fourier transform and inverse fast Fourier transform; ZP stands for zero-padding. One usually obtains superior reconstructions when some smoothing is also incorporated in (68). Smoothing may be implemented by multiplying the product of the two FFTs by a Hamming window. When such a window is incorporated, (68) may be rewritten as

After the filtered projections Q (nt) are calculated with the alternative method presented here, the rest of the implementation for reconstructing the image is the same as in the preceding subsection. That is, we use (45) for backprojections and their summation. Again for a given (x, y) and the argument x cos + y sin may not correspond to one of the kt at which Q i is known. This will call for interpolation and often linear interpolation is adequate. Sometimes, in order to eliminate the computations required for interpolation, preinterpolation of the functions Q (t) is also used. In this technique, which can be combined with the computation in (69), prior to backprojection, the function Q (t) is preinterpolated onto 10 to 1000 times the number of points in the projection data. From this dense set of points one simply retains the nearest neighbor to obtain the value of Q i at x cos , + y sin j. A variety of techniques are available for preinterpolation [Sch73]. One method of preinterpolation, which combines it with the operations in (69), consists of the following: In (69), prior to performing the IFFT, the frequency domain function is padded with a large number of zeros. The inverse transform of this sequency yields the preinterpolated Q . It was recently shown [Kea78] that if the data sequence contains "fractional" frequencies this approach may lead to large errors especially near the beginning and the end of the data sequence. Note that with preinterpolation and with appropriate programming, the backprojection for parallel projection data can be accomplished with virtually no multiplications. Using the implementation in (68), Fig. 3.17(b) shows the reconstructed values on the line y = — 0.605 for the Shepp and Logan head phantom. Comparing with Fig. 3.13(b), we see that the dc shift and the dishing have been eliminated. Fig. 3.17(a) shows the complete reconstruction. The number of rays used in each projection was 127 and the number of projections 100. To make convolutions aperiodic, the projection data were padded with zeros to make each projection 256 elements long.

3.4 Reconstruction from Fan Projections The theory in the preceding subsections dealt with reconstructing images from their parallel projections such as those shown in Fig. 3.1. In generating these parallel data a source-detector combination has to linearly scan over the

ALGORITHMS FOR RECONSTRUCTION WITH NONDIFFRACTING SOURCES

75

1-2

DESCRIBING THE PROBABILITY LAW OF A STOCHASTIC PROCESS

25

Fig. 1.2. Typical sample functions of the stochastic process X(t) = R cos (ut + &).

where co is a known positive constant, and A and B are independent normally distributed random variables with means 0 and variances o-2; where u is a known positive constant, R and & are independent random variables, 6 is uniformly distributed on 0 to 2ir, and R is Rayleigh distributed with probability density function for r > 0

2B Two stochastic processes {X(t), t > 0} and {Y(t), t > 0} are said to be independent if, for any integer n and n time points ti, • • • , tn, the random vectors (J(ti), • • • , *(*.)) and

(y(«i),---,y(fcO)

are independent in the sense that for all (Borel) sets A and B of n-tuples of real numbers

Show that the processes defined by Eqs. 2.8 and 2.9 are independent if the random variables A, B, R, and 6 are independent. 2C Show that if one specifies the n-dimensional probability laws of a stochastic process (that is, the set of all n-dimensional distributions) one thereby specifies all lower dimensional probability laws of the process. Hint. Show that for any set {fi, • • • , tn] and m < n,

26

RANDOM VARIABLES AND STOCHASTIC PROCESSES

Ch 1

EXERCISES In Exercises 2.1-2.3, consider a sequence of independent random variables Xi, Xt, • • • identically distributed as a random variable X. Let Sn = Xi + Xi -\ f- Xn. Under the assumption given in each exercise concerning the probability law of X, find for any integer n, (i) the joint characteristic function of X\, • • • , Xn; (ii) the joint characteristic function of 5i, • • • , Sn; (iii) the joint characteristic function of YI, • • •, Yn, where Y k — Xk+\ — X),. 2.1 X is AT(0,ff*). 23, X is Poisson distributed with mean X. 24 X is exponentially distributed with mean 1A-

In Exercises 2.4 — 2.8 a stochastic process {X(l), t > 0} is denned by an explicit formula in terms of two random variables A and B, which are independent normally distributed random variables with mean 0 and variance a 2 . Find the mean of the following random variables:

i n TO

THE WIENER PROCESS AND THE POISSON PROCESS

Two stochastic processes, the Wiener process and the Poisson process, play a central role in the theory of stochastic processes. These processes are valuable, not only as models for many important phenomena, but also as building blocks with which other useful stochastic processes can be constructed. In order to define these processes, we must introduce the notion of a stochastic process with independent increments.

1-3

THE WIENER PROCESS AND THE FOISSON PROCESS

27

A continuous parameter stochastic process {X((), 0 < t < } is said to have independent increments if Jf (0) = 0 and, for all choices of indices 4> < h < • • • < £», the n random variables

are independent. The process is said to have stationary independent increments if, in addition, X(t 2 -+• h) — X(ti -f h) has the same distribution as X(ti) — X(l{) for all choice of indices ti and < 2 , and every h > 0. It is easily verified, for a stochastic process with independent increments, that

Consequently, from a knowledge of the individual probability laws of X(t) and X(t) — X(s), for all non-negative s and t, one can deduce the joint probability law of any n random variables X(ti), • • • , A"(J«). The Wiener process. In the theory of stochastic processes and its applications, a fundamental role is played by the Wiener process. Among other applications, the Wiener process provides a model for Brownian motion and thermal noise in electric circuits. A particle (of, say, diameter 10~4 cm) immersed in a liquid or gas exhibits ceaseless irregular motions, which are discernible under the microscope. The motion of such a particle is called Brownian motion, after the English botanist Robert Brown, who discovered the phenomenon in 1827 (just after the introduction of microscopes with achromatic objectives; see Perrin [1916], p. 84). The same phenomenon is also exhibited in striking fashion by smoke particles suspended in air. The explanation of the phenomenon of Brownian motion was one of the major successes of statistical mechanics and kinetic theory. In 1905, Einstein showed that Brownian motion could be explained by assuming that the particles are subject to the continual bombardment of the molecules of the surrounding medium. His pioneering work was generalized, extended, and experimentally verified by various physicists. (For a history of the theory of Brownian motion, see the papers collected in Wax [1954] and Einstein [1956].) Let X(t) denote the displacement (from its starting point) after time t of a particle in Brownian motion. By definition X(0) = 0. A Brownian particle undergoes perpetual motion due to continual "impacts" upon the particle due to the force field of its surrounding medium. The

28

RANDOM VARIABLES AND STOCHASTIC PROCESSES

Ch 1

displacement of the particle over a time interval (s,t) which is long compared to the time between impacts can be regarded as the sum of a large number of small displacements. Consequently by the central limit theorem it seems reasonable to assume that X(t) — X(s) is normally distributed. Next, it is reasonable to assume that the probability distribution of the displacement X(f) — X(s) should be the same as that of X(t + h) — X (s + h), for any h > 0, since it is supposed that the medium is in equilibrium and the probability law of a particle's displacement over an interval should depend only on the length t — s of the interval, and not on the time at which we begin observation. We next assume that the motion of the particle is due entirely to very frequent irregular molecular impacts. Mathematically, this is interpreted to say that the stochastic process X(t) has independent increments; the displacements over non-overlapping intervals are independent since the number and size of impacts in non-overlapping intervals are independent. We are thus led to define the notion of a Wiener process (after N. Wiener who was among the first to consider mathematical Brownian motion [1923], [1930]). The Wiener process is also called the Wiener-Levy process, and the Brownian motion process. A stochastic process {X(t), t > 0} is said to be a Wiener process if (i) {X(t), t > 0} has stationary independent increments, (ii) For every t > 0, X(i) is normally distributed, (iii) For all t > 0, E[X(£)] = 0, (iv) X(0) = 0. Since X(t) has independent increments, and X(0) = 0, to state the probability law of the stochastic process X(t) it suffices to state the probability law of the increment X(t) — X(s) for any s < t. Since X(t) — X(s) is normal its probability law is determined by its mean and variance. One easily verifies that E[X(s) - X(t)] = 0. Therefore,

From the fact that X(f) has stationary independent increments it may be shown (see Exercise 3.15) that there is some positive constant, denoted by a\ such that for t > s > Q

The probability law of a Wiener process is thus determined by axioms (i)-(iv) up to a parameter 0} constitutes a stochastic process. For each random variable N ( t ) , the only possible values are the integers 0, 1, 2, • • • . A stochastic process (N(f), t > 0} whose random variables can assume as values only the integers 0, 1, 2, • • • is called an integer-valued process. A particularly important integer-valued process is the Poisson process.

30

RANDOM VARIABLES AND STOCHASTIC PROCESSES

Ch 1

Fig. 1.3. Typical sample function of a Poisson process {N(f), t > 0} with mean rate v = 1 per unit time.

An integer-valued process {N(t), t > 0} is said to be a Poisson process, with mean rate (or intensity) v, if the following assumptions are fulfilled: (i) {N(t), t > 0} has stationary independent increments; (ii) for any times 5 and t such that s < t the number N(f) — N(s) of counts in the interval « to t is Poisson distributed, with mean v(t — s). Consequently, for k = 0, 1, 2, • • •

From Eq. 3.7 one sees that the parameter v represents the mean rate of occurrence per unit time of the events being counted. There are several very general sets of assumptions which lead to the Poisson process. These are stated in Chapter 4. It will be seen that the conditions which an integer-valued stochastic process must satisfy in order to be a Poisson process may be expected to arise often. Events whose counting function N( • ) is a Poisson process will be said to be occurring in accord with a Poisson process at a mean rate v or to be of Poisson type with intensity v.

1-3

THE WIENER PROCESS AND THE POI86ON PROCESS

31

EXAMPLE 3A Radioactive decay. All modern theories of radioactive decay involve the assumption that in an assembly of nuclei of a given element all the nuclei are identical, independent, and have the same probability of decaying in unit time. It then follows that emissions of radioactive radiations from a radioactive source constitute events whose counting process is a Poisson process. An example of an experimental verification of the fact that the Poisson process adequately describes radioactive disintegration is given in Evans (1955), p. 777 ff. EXAMPLE SB Shot noise in electron tabes. The sensitivity attainable with electronic amplifiers and apparatus is inherently limited by the spontaneous current fluctuations present in such devices, usually called noise. One source of noise in vacuum tubes is shot noise, which is due to the random emission of electrons from the heated cathode. Suppose that the potential difference between the cathode and the anode is so great that all electrons emitted by the cathode have such high velocities that there is no accumulation of electrons between the cathode and the anode (and thus no space charge). Emission of electrons from the cathode may be shown to be events of Poisson type (see Davenport and Root [1958], pp. 112-119). Consequently, the number of electrons emitted from the cathode in a time interval of length t obeys a Poisson probability law with parameter vt, in which r is the mean rate of emission of electrons from the cathode. EXAMPLE 3C Machine failures. Consider an item (such as a vacuum tube or a machine) which is used until it fails and is then repaired (and consequently renewed) or replaced by a new item of similar type. The lifetime (length of time such an item serves before it fails) is assumed to be a random variable T. The lifetimes Ti, Tiy • • •, T*, • • • of the successive items put into service are assumed to be independent random variables, identically distributed as T. For t > 0, let N(() be the number of items that have failed in the interval 0 to t. If the lifetime of each item is exponentially distributed, it may be shown that {N(t)t t > 0} is a Poisson process. Indeed it is shown in Chapters 4 and 5 that (N(t), t > 0} is a Poisson process if and only if each item has an exponentially distributed lifetime. Other examples of events of Poisson type, and methods of determining whether a given process is Poisson, may be found in Davis (1952). Poisson processes in space. In developing theories of the distribution of galaxies in the stellar system or of the distribution of centers of popu-

32

RANDOM VARIABLES AND STOCHASTIC PROCESSES

Ch 1

lations (animals, epidemics, and so on) it is convenient to regard the centers of the galaxies or the populations as points distributed in space. Consider an array of points distributed in a space S, where S is a Euclidean space of dimension d > 1. For each region R in S, let N(R) denote the number of points (finite or infinite) contained in the region R. The array of points is said to be distributed according to a stochastic mechanism if for every (measurable) region R in S, N(R) is a random variable. The array of points is said to be of Poisson type with density v or to be distributed in accord with a Poisson process with density v if the following assumptions are fulfilled: (i) the numbers o f points in non-overlapping regions are independent random variables; more precisely, for any integer n and n non-overlapping (disjoint) regions R i, R 2, • • • , Rn, the random variables N(R i), • • • , N(R^) are independent; (ii) for any region R of finite volume, N(R) is Poisson distributed with mean vV(R), where V(R) denotes the ((^-dimensional Euclidean) volume of the region R. EXAMPLE 3D Spatial distribution of plants and animals (ecology). The distributions in space of animals and plants are described by Poisson processes and generalized Poisson processes (denned in Section 4-2). The reason for this is well described by Skellam (1952): "Consider a wide expanse of exposed open ground of a uniform character such as would be provided by the muddy bed of a recently drained shallow lake, and consider the disposition of the independently dispersed windborne seeds of one of the species [of plants] which will colonize the area. That the number occurring in a quadrat square marked on the surface is a Poisson variate is seen from the fact that there are many such seeds each with an extremely small chance of falling into the quadrat." If seeds are distributed on a plane in accord with a Poisson process at a mean rate v per unit area, then the number of seeds in a region of area A has a Poisson distribution with mean vA. To illustrate the usefulness of the Poisson process in space, we obtain the following result. THEOREM 3A Distribution of the nearest neighbor in a Poisson distribution of particles in space, (i) If particles are distributed on a plane in accord with a Poisson process at a mean rate v per unit area, and T is the distance between an individual particle and its nearest neighbor, then mT- is exponentially distributed with mean 1. (ii) If particles are distributed in threedimensional space in accord with a Poisson process at a mean rate v per unit

1-3

THE WIENER PROCESS AND THE POISSON PROCESS

33

volume, and if T is the distance between an individual particle and its nearest neighbor, then (4/3)in'T3 is exponentially distributed with mean 1; consequently, for t > 0,

Remark. Eq. 3.8 plays an important role in astrophysics in the analysis of the force acting on a star which is a member of a stellar system (see Chandrasekhar [1943], p. 72), Proof. We explicitly prove only assertion (i). First note that P[T2 > y/m>] is equal to the probability that a circle of radius {y/in>} 1/2 contains zero particles. Now the probability that a circle of area A contains zero particles is given by e~"A. Since a circle of radius {y/m>} l!Z has area y/v, it follows that

and the desired conclusion is at hand. EXERCISES Solve Exercises 3.1 to 3.9 by defining a stochastic process {N(t}, t > 0} with stationary independent increments in terms of which one may formulate the problem. 3.1 If customers arrive at the rate of 2 a minute, what is the probability that the number of customers arriving in 2 minutes is (i) exactly 3, (ii) 3 or less, (iii) 3 or more, (iv) more than 3, (v) less than 3? 3.2 If customers arrive at the rate of 2 a minute, what is the probability that in each of two non-overlapping 2-minute periods, the number of customers arriving is (i) exactly 3, (ii) 3 or less, (iii) 3 or more? 3.3 If customers arrive at the rate of 2 a minute, find (i) the mean number of customers in a 5-minute period, (ii) the variance of the number of customers in a 5-minute period, (iii) the probability that there will be at least 1 customer in a 5-minute period. 3.4 Certain parts essential to the running of a machine fail in accord with a Poisson process at a rate of 1 every 5 weeks. If there are 2 spare parts in inventory, and if a new supply will arrive in 9 weeks, what is the probability that during the next 9 weeks, due to a lack of spare parts, production will be stopped for a week or more? 3.5 What is the probability that a stock of 4 items of a certain product will last less than a day if sales of the item constitute (i) a Poisson process with an average daily volume of 4 units, (ii) a Poisson process whose average daily volume is a random variable equal to 3, 4, 5 with probabilities 0.25, 0.50, and 0.25 respectively?

34

RANDOM VARIABLES AND STOCHASTIC PROCESSES

Ch 1

3.6 The customers of a certain newsboy arrive in accord with a Poisson probability law at a rate of 1 customer per minute. What is the probability that 5 minutes have elapsed since (i) his last customer arrived, (ii) his next to last customer arrived? 3.7 The customers of a newsboy arrive in accordance with a Poisson process, at a mean arrival rate of 2 a minute, (i) Find the conditional probability that there will be no customers within the next 2 minutes, given that there were 1 or more customers within the last 2 minutes, (ii) The newsboy often makes the following bet. He will pay his opponent $1 if his next customer does not arrive within a minute; otherwise the opponent must pay him $1. What is the expected value of the newsboy's winnings on this bet? 3.8 A radioactive source is observed during 4 non-overlapping time intervals of 6 seconds each. The number of particles emitted during each period is counted. If the particles emitted obey a Poisson probability law, at a rate of 0.5 particles emitted per second, find the probability that (i) in each of the 4 time intervals, 3 or more particles are counted, (ii) in at least 1 of the 4 time intervals, 3 or more particles are counted. 3.9 Consider the number of suicides in a certain city in which suicides occur at an average rate of 2 per week, (i) Find the probability that in the town there will be 6 or more suicides in a week, (ii) What is the expected number of weeks in a year in which 6 or more suicides will be reported in this town? (iii) Would you find it surprising that during 1 year there were at least 2 weeks in which 6 or more suicides were reported? (iv) If only every other suicide is reported, what is the probability that in a week 2 or more suicides are reported? 3.10 Prove assertion (ii) of Theorem 3A. 3.11 Find the probability density function of the distribution of the nearest neighbor in a Poisson distribution of particles in d-dimensional Euclidean space. 3.12 Show that the mean distance E[T] between a particle and its nearest neighbor in a collection of particles randomly distributed in 3-dimensional space is given by

3.13 Consider events happening on the interval — « to « in accord with a Poisson process with density v. For each time t > 0, let Ut be the time from the most recent event happening before t to t, and let V< be the time from t to the first event happening after t. Find the probability density functions of Vt and Ut + Vt. 3.14 Let {Tn} be a sequence of independent random variables exponentially distributed with mean l/v. For t > 0, define N(t) as the integer satisfying

1-4

3.15

TWO-VALUED PROCESSES

35

Show that {N(t), t > 0} is a Poisson process with mean rate v. Note. This assertion is proved in Theorem 2B of Chapter 5. Prove that Eq. 3.4 holds. Hint. The function f(t) = Var[X"(fl] satisfies the functional equation discussed in Section 4-1.

3.16

Let {X(t), t > 0} be a Poisson process. Show that

In view of this result would you consider X(t)/t to be a satisfactory estimate of v on the basis of having observed the process X(f) over the interval Oto