Question

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)=αβX^β-1 exp(-αX^β)

a.      Derive 98% large sample confidence interval for α.

b.      Find maximum likelihood estimator of α.

Answer 1

Answer:

Given Data

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

Here β=1.

which is the pdf of exponential distribution

a)  Derive 98% large sample confidence interval for α.

98% confidence interval for α is

Let Y = 2 α x

Let

be the percentile and percentile.

Then b)    Find maximum likelihood estimator of α.

MLE of α is -

Likelihood function is given by -

Log . likelihood function is given by-

Differentiating w.r.t  α  we have ,

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