A manufacturing company produces many differentproducts for wholesale distribution. Its flagship product,a reverse cylindrical dipolar tube, has a failure distributiongiven by the following PDF:f(t) = {0.000192t2 0sts 25 yearselsewhere(a) Find R(15) and MTTF.(b) Find the probability that a failure will occur after thesecond year and by the end of the tenth year.(c) Find the median time to failure.(d) Construct the hazard rate function, At. Is it IFR, DRF,or CFR?

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Answered: A manufacturing company produces many…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Consider randomly selecting n segments of pipe and determining the corrosion loss (mm) in the wall thickness for each one. Denote these corrosion losses by Y₁' Yn. The article "A Probabilistic Model for a Gas Explosion Due to Leakages in the Grey Cast Iron Gas Mains"+ proposes a linear corrosion model: Y; = t;R, where t; is the age of the pipe and R, the corrosion rate, is exponentially distributed with parameter 1. Obtain the maximum likelihood estimator of the exponential parameter (the resulting mle appears in the cited article). [Hint: If c> 0 and X has an exponential distribution, so does cX.] O Â = O Â = O λ = * * O n Srit) i = 1 n j = 1 Σ i = 1 n n i = 1 n LY i = 1 n 0 1 = L (rt) 2 i = 1 | = 1 n n Y; | = 1
A CI is desired for the true average stray-load loss μ (watts) for a certain type of induction motor when the line current is held at 10 amps for a speed of 1500 rpm. Assume that stray-load loss is normally distributed with = 2.1. (Round your answers to two decimal places.) (a) Compute a 95% CI for μ when n = 25 and x = 56.0. watts (b) Compute a 95% CI for μ when n = 100 and x = 56.0. watts (c) Compute a 99% CI for μ when n = 100 and x = 56.0. watts (d) Compute an 82% CI for μ when n = 100 and x = 56.0. watts (e) How large must n be if the width of the 99% interval for u is to be 1.0? (Round your answer up to the nearest whole number.) n =
Consider an item with time to failure T that has a Weibull distribution with shape parameter 2.25 and scale parameter 0.0004 per hours. When the item fails it is repaired to an as good as new condition. The repairtime (downtime) D has a lognormal distribution with parameters 1.5 and 0.42. The item is supposed to be in continuous operation. (a) Find the average availability of the item. (b) A preventive maintenance task that takes 5 hours is performed every 300 hours. Find the operational availability of the item.
What was the maximum sustained yield under the logistic harvest model when k=500 and r=0.4? Values 1 500 1000 49L K L N N.d num [1:601] 0 123456789... num [1:50] 5 157.502 1079.678 0.732 24.054 ... num [1:50] 114 329 349 350 350 ... 0.4 int [1:50] 1 2 3 45678910 50 t Nt r Z Z L P T time y Files Plots Packages Help Viewer Presentation Zoom Export num [1:601] 0.4 0.399 0.398 0.398 0.397 ... Publish Observed Growth Rate 69 -0.1 0.1 0.3 0.5 Observed Growth Maximum Growth Carrying Capacity 0 100 200 300 400 500 600
DHL usually gives its customer a 60-minute window in which it expects home deliveries to be delivered. An independent consultant estimates that the final delivery time within the window is a delivery that follows the following condensation function:a. Draw the density function of the event. b. What is the expected value of the transaction? c. What is the variance of the event?
Suppose that the interval between eruptions of a particular geyser can be modelled by an exponential distribution with an unknown parameter 0 > 0. The probability density function of this distribution is given by f(x; 0) = 0e 0¹, x > 0. The four most recent intervals between eruptions (in minutes) are x₁ = 32, x₂ = 10, x3 = 28, x4 = 60; their values are to be treated as a random sample from the exponential distribution. (a) Show that the likelihood of based on these data is given by L(0) 04-1306 = (b) Show that L'(0) is of the form L'(0) = 0³ e 1300 (4- 1300). (c) Show that the maximum likelihood estimate of 0 based on the data is ~ 0.0308 making your argument clear. (d) Explain in detail how the maximum likelihood estimate of that you have just obtained in part (c) relates to the maximum likelihood estimator of for an exponential distribution.
Consider independent observations y₁, ..., yn from the model Y₁ likelihood 1(µ) as appropriate, compute the following items. 1. Derive the maximum likelihood estimate û. 2. Write the second derivative of log-likelihood 1(μ). Poisson(μ). Using likelihood L(µ) and log-
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2.11 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.
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