Concept explainers
During an eight-hour shift, the proportion of time Y that a sheet-metal stamping machine is down for maintenance or repairs has a beta distribution with α = 1 and β = 2. That is,
The cost (in hundreds of dollars) of this downtime, due to lost production and cost of maintenance and repair, is given by C = 10 + 20Y + 4Y2. Find the
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Chapter 4 Solutions
Mathematical Statistics with Applications
- Let Y1 < Y2 < · · · < Yn be the order statistics of a random sample of size nfrom a distribution with pdf f(x) = 1, 0 < x < 1, zero elsewhere. Show that thekth order statistic Yk has a beta pdf with parameters α = k and β = n − k + 1.arrow_forwardI need helping finding the Jeffreys’ prior for the parameter alpha of the Maxwell distribution and then finding a transformation of this parameter in which the corresponding prior is uniform.arrow_forwardConsider a random sample X1,...,Xn (n > 2) from Beta(θ,1), where we wish to estimate the parameter θ. (a) Find the MLE θˆ and write it as a function of T = − ∑ni=1 log Xi. (b) Find the sampling distribution of T = − ∑ni=1 log Xi . (Hint: First find the distribution of Ti = − log Xi .)arrow_forward
- Let X denote 0.025 × the ambient air temperature (˚C) and let Y denote the time (min) that it takes for a diesel engine to warm up. Assume that (X, Y) has joint probability density function f(x,y) = 1.6x (1 − x)(6 + 5x − 4y), for 0 < x < 1, 0 < y < 0.5. While you cannot guess the value of the correlation from the regression curve for X or Y, do they suggest whether it likely is positive or negative?arrow_forwardSuppose that X is a continuous unknown all of whose values are between -3 and 3 and whose PDF, denoted f , is given by f ( x ) = c ( 9 − x^2 ) , − 3 ≤ x ≤ 3 , and where c is a positive normalizing constant. What is the variance of X?arrow_forwardFind the maximum likelihood estimator for θ in the pdf f(y; θ) = 2y/(1 − θ^2), θ ≤ y ≤ 1.arrow_forward
- A researcher is interested in testing whether annual house hold income in Philadelphia is normal. So she took a sample of 50 house holds and found that skewness (s)= 2.3190 and Kurtosis (k) = 6.7322. Use the Jarque- Bera Test to test , at alpha 0.05, whether income follows normal distribution. -Yes, population is normal because Chi-Square test is higher than critical value. -Yes, population is normal because Chi-Square test is less than critical value. -No, population is not normal because Chi-Square test is higher than critical value. -No, population is not normal because Chi-Square test is less than critical value.arrow_forwardFor a certain psychiatric clinic suppose that the random variable X represents the total time (in minutes) that a typical patient spends in this clinic during a typical visit (where this total time is the sum of the waiting time and the treatment time), and that the random variable Y represents the waiting time (in minutes) that a typical patient spends in the waiting room before starting treatment with a psychiatrist. Further, suppose that X and Y can be assumed to follow the bivariate density function fXY(x,y)=λ2e−λx, 0<y<x, where λ > 0 is a known parameter value. (a) Find the marginal density fX(x) for the total amount of time spent at the clinic. (b) Find the conditional density for waiting time, given the total time. (c) Find P (Y > 20 | X = x), the probability a patient waits more than 20 minutes if their total clinic visit is x minutes. (Hint: you will need to consider two cases, if x < 20 and if x ≥ 20.)arrow_forward
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