Question

In: Statistics and Probability

The lifespan of a system component follows a Weibull distribution with α (unknown) and β=1. (hint:...

The lifespan of a system component follows a Weibull distribution with α (unknown) and β=1. (hint: start with a confidence interval for μ)

f(x)=αβXβ-1 exp(-αXβ)

a.      Derive 92% large sample confidence interval for α.

b.      Find the maximum likelihood estimator of α.

Solutions

Expert Solution

Solution:-

Given that

The probability density function of lifespan of system component is given by

Here,

which is the pdf of exponential Distribution.

a) Derive 92% large sample confidence interval for .

92% confidence Interval for is

Let

Let and be the th percentile and 51 th percentile.

Then

The 92% confidence Interval is

b)  Find the maximum likelihood estimator of .

MLE of is

likelihood function is given by

Log likelihood function is given by

Differentiating w.r.t we have,

MLE  

Thanks for supporting...

Please give positive rating...

orchestra answered 1 year ago
Next > < Previous

Related Solutions

The lifespan of a system component follows a Weibull distribution with α (unknown) and β=1. (hint:...
The lifespan of a system component follows a Weibull distribution with α (unknown) and β=1. (hint: start with confidence interval for μ) f(x)=αβXβ-1 exp(-αXβ) a.      Derive 98% large sample confidence interval for α. b.      Find maximum likelihood estimator of α.
The lifespan of a system component follows a Gamma distribution with β (unknown) and α=1. (hint:...
The lifespan of a system component follows a Gamma distribution with β (unknown) and α=1. (hint: start with confidence interval for μ) f(x)=(1/βα Γ(α)) Xα-1 e-X/β a.      Derive 98% large sample confidence interval for β. b.      Find maximum likelihood estimator of β. a) 98% large sample confidence interval fo β is b) MLE for β is
The lifetime of an electronic component has a Weibull distribution with parameters α​=0.60 and β​=2. Compute...
The lifetime of an electronic component has a Weibull distribution with parameters α​=0.60 and β​=2. Compute the probability the component fails before the expiration of a 4 year warranty. The maximum flood​ levels, in millions of cubic feet per​ second, for a particular U.S. river have a Weibull distribution with α=5/3 and β=3/2. Find the probability that the maximum flood level for next year will exceed 0.7 million cubic feet per second.
Assume the life of a roller bearing follows a Weibull distribution with parameters β=2 and δ=7,500...
Assume the life of a roller bearing follows a Weibull distribution with parameters β=2 and δ=7,500 hours. Determine the probability that the bearing will last at least 8000 hours. Verify your answer using R. Determine the expect value and the variance.
Assume the life of a roller bearing follows a Weibull distribution with parameters β=2 and δ=7,500...
Assume the life of a roller bearing follows a Weibull distribution with parameters β=2 and δ=7,500 hours. Determine the probability that the bearing will last at least 8000 hours. Determine the expect value and the variance.
Suppose X follows a Gamma distribution with parameters α, β, and the following density function f(x)=...
Suppose X follows a Gamma distribution with parameters α, β, and the following density function f(x)= [x^(α−1)e^(−x/ β)]/ Γ(α)β^α . Find α and β so that E(X)= Var(X)=1. Also find the median for the random variable, X.
Suppose a car radiator useful lifetime has Weibull distribution with β = 1.5 and the mean...
Suppose a car radiator useful lifetime has Weibull distribution with β = 1.5 and the mean lifetime of 150,000 miles. John has just bought a used car with 180,000 miles and original radiator which is still good. What is the probability that it’s going to last at least 20,000 miles more before needing replacement?
Let α, β be cuts as defined by the following: 1) α ≠ ∅ and α...
Let α, β be cuts as defined by the following: 1) α ≠ ∅ and α ≠ Q 2) if r ∈ α and s ∈ Q satisfies s < r, then s ∈ α. 3) if r ∈ α, then there exists s ∈ Q with s > r and s ∈ α. Let α + β = {r + s | r ∈ α and s ∈ β}. Show that the set of all cuts R with the...
The lifespan of an electrical component has an exponential distribution with parameter lambda = 0.013. Suppose...
The lifespan of an electrical component has an exponential distribution with parameter lambda = 0.013. Suppose we have an iid sample of size 100 of these components Some hints: P(X < c) for an exponential(lambda) can be found via pexp(c,lambda) E[X] = 1/lambda and Var[X] = 1/lambda^2 Round all answers to 4 decimals Using the exact probability distribution, what is the probability that a single component will be within 15.38 units of the population mean? Using Chebyshev's inequality, what is...
1. Homework 4 Due: 2/28 Let X1,...,Xn i.i.d. Gamma(α,β) with α > 0, β > 0...
1. Homework 4 Due: 2/28 Let X1,...,Xn i.i.d. Gamma(α,β) with α > 0, β > 0 (a) Assume both α and β are unknown, find their momthod of moment estimators: αˆMOM and βˆMOM. (b) Assume α is known and β is unknown, find the maximum likelihood estimation for β. 2.
ADVERTISEMENT
Subjects
ADVERTISEMENT
Latest Questions
ADVERTISEMENT