Mathematical Methods in the Physical Sciences

3rd Edition

ISBN: 9780471198260

Author: Mary L. Boas

Publisher: Wiley, John & Sons, Incorporated

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Textbook Question

Chapter 8.10, Problem 3P

Use the convolution integral to find the inverse transforms of:

p p 2 − 1 2 = p p 2 − 1 ⋅ 1 p 2 − 1

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Chapter 8.10, Problem 2P

Chapter 8.10, Problem 4P

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Module Code: MATH380202 3. (a) Let {} be a white noise process with variance σ2. Define an ARMA(p,q) process {X} in terms of {+} and state (without proof) conditions for {X} to be (i) weakly stationary and (ii) invertible. Define what is meant by an ARIMA (p, d, q) process. Let {Y} be such an ARIMA(p, d, q) process and show how it can also be represented as an ARMA process, giving the AR and MA orders of this representation. (b) The following tables show the first nine sample autocorrelations and partial auto- correlations of X and Y₁ = VX+ for a series of n = 1095 observations. (Notice that the notation in this part has no relationship with the notation in part (a) of this question.) Identify a model for this time series and obtain preliminary estimates for the pa- rameters of your model. X₁ = 15.51, s² = 317.43. k 1 2 3 4 5 6 7 Pk 0.981 0.974 0.968 akk 0.981 0.327 8 9 0.927 0.963 0.957 0.951 0.943 0.935 0.121 0.104 0.000 0.014 -0.067 -0.068 -0.012 Y₁ = VX : y = 0.03, s² = 11.48. k 1…

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Module Code: MATH380202 1. (a) Define the terms "strongly stationary" and "weakly stationary". Let {X} be a stochastic process defined for all t € Z. Assuming that {X+} is weakly stationary, define the autocorrelation function (acf) Pk, for lag k. What conditions must a process {X+) satisfy for it to be white noise? (b) Let N(0, 1) for t€ Z, with the {+} being mutually independent. Which of the following processes {X+} are weakly stationary for t> 0? Briefly justify your answers. i. Xt for all > 0. ii. Xo~N(0,) and X₁ = 2X+-1+ &t for t > 0. (c) Provide an expression for estimating the autocovariance function for a sample X1,..., X believed to be from a weakly stationary process. How is the autocor- relation function Pk then estimated, and a correlogram (or acf plot) constructed? (d) Consider the weakly stationary stochastic process ✗+ = + + +-1+ +-2 where {E} is a white noise process with variance 1. Compute the population autocorre- lation function Pk for all k = 0, 1, ....

Chapter 8 Solutions

Mathematical Methods in the Physical Sciences

Ch. 8.1 - Verify the statement of Example 2. Also verify...Ch. 8.1 - Solve Example 4 using the general solution...Ch. 8.1 - Verify that y=sinx,y=cosx,y=eix, and y=eix are all...Ch. 8.1 - Find the distance which an object moves in time t...Ch. 8.1 - Find the position x of a particle at time t if its...Ch. 8.1 - A substance evaporates at a rate proportional to...Ch. 8.1 - The momentum p of an electron at speed v near the...Ch. 8.2 - For each of the following differential equations,...Ch. 8.2 - For each of the following differential equations,...Ch. 8.2 - For each of the following differential equations,...

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