Mathematics For Machine Technology
8th Edition
ISBN: 9781337798310
Author: Peterson, John.
Publisher: Cengage Learning,
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Chapter 34, Problem 29A
To determine
Find the readings in the table from the help of given data.
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3. Let {X} be an autoregressive process of order one, usually written as AR(1).
(a) Write down an equation defining X₁ in terms of an autoregression coefficient a
and a white noise process {} with variance σ².
Explain what the phrase "{} is a white noise process with variance o?" means.
(b) Derive expressions for the variance 70 and the autocorrelation function Pk, k
0,1,. of the {X} in terms of o2 and a.
Use these expressions to suggest an estimate of a in terms of the sample autocor-
relations {k}.
(c) Suppose that only every second value of X is observed, resulting in a time series
Y X2, t = 1, 2,....
Show that {Y} forms an AR(1) process. Find its autoregression coefficient, say
d', and the variance of the underlying white noise process, in terms of a and o².
(d) Given a time series data set X1, ..., X256 with sample mean = 9.23 and sample
autocorrelations ₁ = -0.6, 2 = 0.36, 3 = -0.22, p = 0.13, 5 = -0.08,
estimate the autoregression coefficients a and a' of {X} and {Y}.
Refer to page 96 for a problem involving the heat equation. Solve the PDE using the method of
separation of variables. Derive the solution step-by-step, including the boundary conditions.
Instructions: Stick to solving the heat equation. Show all intermediate steps, including separation
of variables, solving for eigenvalues, and constructing the solution. Irrelevant explanations are
not allowed.
Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440AZF/view?usp=sharing]
Refer to page 83 for a vector field problem requiring verification of conservative nature and
finding a scalar potential function.
Instructions: Focus strictly on verifying conditions for conservativeness and solving for the
potential function. Show all work step-by-step.
Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440AZF/view?usp=sharing]
Chapter 34 Solutions
Mathematics For Machine Technology
Ch. 34 - Use an electronic vernier caliper to measure the...Ch. 34 - Read the metric vernier depth gage measurement for...Ch. 34 - Prob. 3ACh. 34 - Prob. 4ACh. 34 - Prob. 5ACh. 34 - Prob. 6ACh. 34 - Prob. 7ACh. 34 - Read the settings on the following 0.001-inch...Ch. 34 - Prob. 9ACh. 34 - Prob. 10A
Ch. 34 - Prob. 11ACh. 34 - Prob. 12ACh. 34 - Prob. 13ACh. 34 - Prob. 14ACh. 34 - Prob. 15ACh. 34 - Prob. 16ACh. 34 - Prob. 17ACh. 34 - Prob. 18ACh. 34 - Prob. 19ACh. 34 - Prob. 20ACh. 34 - Prob. 21ACh. 34 - Prob. 22ACh. 34 - Prob. 23ACh. 34 - Prob. 24ACh. 34 - Prob. 25ACh. 34 - Prob. 26ACh. 34 - Prob. 27ACh. 34 - Prob. 28ACh. 34 - Prob. 29ACh. 34 - Prob. 30ACh. 34 - Prob. 31ACh. 34 - Prob. 32ACh. 34 - Prob. 33ACh. 34 - Prob. 34ACh. 34 - Prob. 35ACh. 34 - Prob. 36ACh. 34 - Prob. 37ACh. 34 - Prob. 38ACh. 34 - Prob. 39ACh. 34 - Prob. 40ACh. 34 - Prob. 41ACh. 34 - Prob. 42ACh. 34 - Prob. 43ACh. 34 - Prob. 44ACh. 34 - Prob. 45ACh. 34 - Prob. 46ACh. 34 - Prob. 47ACh. 34 - Prob. 48ACh. 34 - Prob. 49ACh. 34 - Read the settings on the following 0.0001-inch...Ch. 34 - Prob. 51ACh. 34 - Prob. 52ACh. 34 - Prob. 53ACh. 34 - Prob. 54ACh. 34 - Prob. 55ACh. 34 - Prob. 56ACh. 34 - Prob. 57ACh. 34 - Prob. 58ACh. 34 - Given the following barrel scale, thimble scale,...Ch. 34 - Prob. 60ACh. 34 - Prob. 61ACh. 34 - Prob. 62ACh. 34 - Prob. 63ACh. 34 - Prob. 64ACh. 34 - Prob. 65ACh. 34 - Prob. 66ACh. 34 - Prob. 67ACh. 34 - Prob. 68ACh. 34 - Prob. 69ACh. 34 - Prob. 70A
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- 1000 1500 2000 Quarterly sales of videos in the Leeds "Disney" store are shown in figure 1. Below is the code and output for an analysis of these data in R, with the sales data stored in the time series object X. Explain what is being done at points (i)-(iv) in the R code. Explain what is the difference between (v) and (vi) in the R code. Explain, giving reasons, which of (v) and (vi) is preferable. Write out the model with estimated parameters in full. (The relevant points in the R code are denoted #2#2#3#23 (i) #### etc.) Given that the sales for the four quarters of 2018 were 721, 935, 649, and 1071, use model-based forecasting to predict sales for the first quarter of 2019. (A point forecast is sufficient; you do not need to calculate a prediction interval.) Suggest one change to the fitted model which would improve the analysis. (You can assume that the choice of stochastic process at (v) in the R code is the correct one for these data.) 2010 2012 2014 Time 2016 Figure 1:…arrow_forward2. Let {X} be a moving average process of order q (usually written as MA(q)) defined on tЄ Z as where {et} is a white noise process with variance 1. (1) (a) Show that for any MA(1) process with B₁ 1 there exists another MA(1) pro- cess with the same autocorrelation function, and find the lag 1 moving average coefficient (say) of this process. (b) For an MA(2) process, equation (1) becomes X=&t+B₁et-1+ B2ɛt-2- (2) i. Define the backshift operator B, and write equation (2) in terms of a polyno- mial function B(B), giving a clear definition of this function. ii. Hence show that equation (2) can be written as an infinite order autoregressive process under certain conditions on B(B), clearly stating these conditions.arrow_forwardRefer to page 92 for a problem involving solving coupled first-order ODEs using Laplace transforms. Instructions: Solve step-by-step using Laplace transforms. Show detailed algebraic manipulations and inversions. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440 AZF/view?usp=sharing] Refer to page 86 for a problem involving solving Legendre's differential equation. Instructions: Solve using power series or standard solutions. Clearly justify every step and avoid unnecessary explanations. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qoHazb9tC440AZF/view?usp=sharing]arrow_forward
- Consider the time series model X₁ = u(t)+s(t) + εt. Assuming the standard notation used in this module, what do each of the terms Xt, u(t), s(t) and & represent? In a plot of X against t, what features would you look for to determine whether the terms μ(t) and s(t) are required? Explain why μ(t) and s(t) are functions of t, whilst t is a subscript in X and εt.arrow_forwardRefer to page 86 for a problem involving solving Legendre's differential equation. Instructions: Solve using power series or standard solutions. Clearly justify every step and avoid unnecessary explanations. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qo Hazb9tC440 AZF/view?usp=sharing] Refer to page 80 for a proof of convergence for a given series using the ratio test. Instructions: Clearly apply the ratio test. Show all steps and provide justification for convergence or divergence. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440AZF/view?usp=sharing]arrow_forwardRefer to page 90 for a problem requiring Fourier series expansion of a given periodic function. Instructions: Clearly outline the process of finding Fourier coefficients. Provide all calculations, integrals, and final expansions. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440 AZF/view?usp=sharing] Refer to page 93 for a problem involving Cauchy-Euler differential equations. Instructions: Solve the given differential equation step-by-step, showing the characteristic roots and general solution clearly.arrow_forward
- Refer to page 80 for a proof of convergence for a given series using the ratio test. Instructions: Clearly apply the ratio test. Show all steps and provide justification for convergence or divergence. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qoHazb9tC440 AZF/view?usp=sharing] Refer to page 94 for a problem requiring the numerical solution of an ODE using the Runge- Kutta method. Instructions: Solve step-by-step, showing iterations, step sizes, and calculations clearly. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qo Hazb9tC440AZF/view?usp=sharing]arrow_forwardRefer to page 82 for a double integral problem. Convert the integral into polar coordinates and evaluate it step-by-step, clearly showing all transformations and limits. Instructions: Focus only on the problem. Provide all steps, including the coordinate transformation, Jacobian factor, and the integral evaluation. Avoid irrelevant details. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440 AZF/view?usp=sharing]arrow_forwardRefer to page 81 for a proof involving the uniqueness of solutions for a given ordinary differential equation. Instructions: Focus strictly on proving the uniqueness theorem using necessary conditions. Justify all intermediate steps. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440AZF/view?usp=sharing]arrow_forward
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