(5 pts) The Exponential distribution, Exp(A), has density function f(z; d)=de-dz for 2, A>0, and is...

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(5 pts) The Exponential distribution, Exp(A), has density function f(z; d)=de-dz for 2, A>0, and is often used for modelling the lifetimes of pieces of equipment. A group of electric motors were tested under extreme conditions. The lifetimes of these motors, in hours, were 3.08, 16.60, 47.24, 9.19, 25.47, 22.29, 12.54, 2.05, 13.32, 2.84 Assume the data are from the Exp(X) distribution. Provide answers to the following to three decimal places. Part a) Find the maximum likelihood estimate of the mean lifetime of the motors. Part b) Find an approximate 95% confidence interval for A, giving both the upper and lower bounds. Lower bound: Upper bound: Part c) Lifetimes are from a "shifted" Exponential distribution if for some >0 the c.d.f. is P(z)=1-e-(2-) for z > L and some > 0. Suppose now the data given above are assumed to be from a shifted Exponential distribution. Evaluate the log-likelihood function for the data when A=0.028 and L = 1.025 Part d) Find the maximum likelihood estimate of L. (5 pts) The Exponential distribution, Exp(A), has density function f(z; d)=de-dz for 2, A>0, and is often used for modelling the lifetimes of pieces of equipment. A group of electric motors were tested under extreme conditions. The lifetimes of these motors, in hours, were 3.08, 16.60, 47.24, 9.19, 25.47, 22.29, 12.54, 2.05, 13.32, 2.84 Assume the data are from the Exp(X) distribution. Provide answers to the following to three decimal places. Part a) Find the maximum likelihood estimate of the mean lifetime of the motors. Part b) Find an approximate 95% confidence interval for A, giving both the upper and lower bounds. Lower bound: Upper bound: Part c) Lifetimes are from a "shifted" Exponential distribution if for some >0 the c.d.f. is P(z)=1-e-(2-) for z > L and some > 0. Suppose now the data given above are assumed to be from a shifted Exponential distribution. Evaluate the log-likelihood function for the data when A=0.028 and L = 1.025 Part d) Find the maximum likelihood estimate of L.

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Part a To find the maximum likelihood estimate MLE of the mean lifetime of the motors we can use the fact that for an Exponential distribution with pa... View the full answer