Concept explainers
Let X denote the vibratory stress (psi) on a wind turbine blade at a particular wind speed in a wind tunnel. The article “Blade Fatigue Life Assessment with Application to VAWTS” (J. of Solar Energy Engr., 1982: 107–111) proposes the Rayleigh distribution, with
as a model for the X distribution.
a. Verify that f (x; θ) is a legitimate pdf.
b. Suppose θ = 100 (a value suggested by a graph in the article). What is the
c. What is the probability that X is between 100 and 200 (again assuming θ = 100)?
d. Give an expression for P(X ≤ x).
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Chapter 4 Solutions
Student Solutions Manual for Devore's Probability and Statistics for Engineering and the Sciences, 9th
- Let X denote the vibratory stress (psi) on a wind turbine blade at a particular wind speed in a wind tunnel. The article, "Blade Fatigue Life AsSsessment with Applications to VAWTS," J. Solar Energy Engr., (1982), pp 107--111, proposes the Rayleigh distribution as a model for the X distribution. Here is the pdf of the model. i e/(2 0), x> 0, f(x, 0) = 0, otherwise. 1 1.1. Suppose that 0= 100 (a value suggested by a graph in the article). What is the probability that X is at most 200? Less that 200? At least 200? 1.2. What is the probability that X is between 100 and 200, assuming 0 = 100, as before? 1 1.3. Determine the expression of the cdf, F(x, 0) = P(Xarrow_forwardA- Locate the cer‘ltroid (X only) of the shaded area Fig.1.Aarrow_forward1. Given that Y = eiax + Cosx, find: dy a. dr b. d'y/dx 2. The ages of a random sample of depositors at a named bank in Lusaka are as follows: 34 28 46 37 33 24 29 45 37 34 32 25 50 54 32 36 38 41 38 44 28 43 40 49 30 46 27 34 61 33 a. Display the data presented above on a stem and leaf display b. State any three reasons why statisticians would display data on a stem and leaf c. Find i. Mean ii. Variance iii. Standard deviation 3. The profits to be realised from airtime vending business, to the nearest S1 000, are believed to follow the probability distribution shown below: -2 000 -1 000 1 000 2 000 3 000 P(X = x) 0.1 0.1 0.2 K 0.3 0.1 a. Determine the value of K. b. Find the probability that the airtime vending i. Makes a loss ii. Realises profit of at least $2 000 c. Find i. The expected earnings and Standard deviation of profits for the air time vending business ii. d. Is the air time vending business likely to be successful? Explain 4. In the table below, 55 students have been…arrow_forwardLet X1,...,Xn ∼ iid Pois(θ), and we wish to estimate φ = Pθ(X = 0) = e−θ. Consider Yi = I (Xi = 0). Show that ∑ni=1 Yi follows a Bin(n, φ) distribution. (d) Using Y1 , . . . , Yn find an unbiased estimator for φ. (e) Is the estimator in (d) the MLE for φ?arrow_forwardIn the vehicle speed test, the results were given in the table below: Find: 1. skew modulus 2. kurtosis coefficient Speed range (km/hr 34-35.9 36-37.9 38-39.9 40-41.9 42-43.9 Frequency (fi) 4 6 4 10 7 Speed range (km/hr 44-45.9 46-47.9 48-49.9 50-51.9 52-53.9 Frequency (fi) 22 8 18 17 16 Speed range (km/hr 54-55.9 56-57.9 58-59.9 60-61.9 62-63.9 Frequency (fi) 13 8 6 4 3arrow_forwardAn article in the Journal of the American Ceramic Society, "Rapid Hot-Pressing of Ultrafine PSZ Powders" (1991, Vol. 74, pp. 1547-1553) considered the microstructure of the ultrafine powder of partially stabilized zirconia as a function of temperature. The data are shown below: ... x = Temperature (°C) | 1100 1200 1300 1100 1500 1200 1300 y = Porosity (%) 30.8 19.2 6 13.5 11.4 7.7 3.6 Find the least squares estimate of the slope. Input answers up to 5 decimal places. Slope = Blank 1 Blank 1 Add your answerarrow_forwardAn article in the Journal of the American Ceramic Society, "Rapid Hot-Pressing of Ultrafıne PSZ Powders" (1991, Vol. 74, pp. 1547-1553) considered the microstructure of the ultrafıne powder of partially stabilized zirconia as a function of temperature. The data are shown below: x= Temperature (°C) 1100 1200 1300 1100 1500 1200 1300 y = Porosity (%) 30.8 11.4 7.7 19.2 6 13.5 3.6 (a) Fit the simple linear regression model using the method of least squares. Find the least squares estimates of the intercept and slope in the simple linear regression model. Find an estimate of o. i (Round your answer to 3 decimal places.) (Round your answer to 5 decimal places.) 6 = i (Round your answer to 2 decimal places.) (b) Estimate the mean porosity for a temperature of 1447 °C. ŷ = (Round your answer to 2 decimal places.)arrow_forwardLet X denote the vibratory stress (psi) on a wind turbine blade at a particular wind speed in a wind tunnel. The article “Blade Fatigue Life Assessment with Application to VAWTS" (J. of Solar Energy Engr., 1982: 107–111) proposes the Rayleigh distribution, with pdf {* -a² /(20²) x > 0 •e f(x; 0) = otherwise as a model for the X distribution. a. Verify that f(x; 0) is a legitimate pdf. b. Suppose 0 = 100 (a value suggested by a graph in the article). What is the probability that X is at most 200? Less than 200? At least 200? c. What is the probability that X is between 100 and 200 (again assuming 0 = 100)? d. Give an expression for P(X < æ).arrow_forward2.11 Let X have the standard normal pdf, fx(x) = (1/√√2π) e¯2²/² (a) Find EX² directly, and then by using the pdf of Y = X2 from Example 2.1.7 and calculating EY. (b) Find the pdf of Y = |X|, and find its mean and variance.arrow_forwardGiven the following p.d.f., f(x) =(x² + x), 0Sx<1 a) Find the E(X) by calculus integration: b) Find E(X²), by calculus integration. c) Find the variance and standard deviation (using findings in parts a) and b) above.arrow_forwardAn article in the Journal of the American Ceramic Society, "Rapid Hot-Pressing of Ultrafine PSZ Powders" (1991, Vol. 74, pp. 1547-1553) considered the microstructure of the ultrafine powder of partially stabilized zirconia as a function of temperature. The data are shown below: ... x= Temperature (°C) 1100 1200 1300 1100 | 1500 1200 1300 y - Porosity (%) 30.8 19.2 6 13.5 11.4 7.7 3.6 Find the least squares estimate of the slope. Input answers up to 5 decimal places. Slope = Blank 1arrow_forwardAn article in the Journal of the American Ceramic Society, "Rapid Hot-Pressing of Ultrafine PSZ Powders" (1991, Vol. 74, pp. 1547-1553) considered the microstructure of the ultrafine powder of partially stabilized zirconia as a function of temperature. The data are shown below: x = Temperature (°C)|1100 1200 1300 1100 1500 1200 1300 y = Porosity (%) 30.8 19.2 6 13.5 11.4 7.7 3.6 Find an estimate of o?arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage