Problems on life testing Assignment 2015

| August 30, 2017

Question
Problems on life testing

Let X1,…..,Xnbe independent distributed with exponential density (2?)-1e-x/2?for x>, and let the ordered X’s be denoted by Y1<2<=…….<n.It is assumed that Y1becomes available first, then Y2, and so on, and that observation is continued until Yrhas been observed. This might arise, for example, in life testing where each X measures the length of life of, say, and electron tube, and n tubes are being tested simultaneously, Another application is to the disintegration of radioactive material, where n is the number of atoms, and observation is continued until r ?-particles have been emitted.
Show that:

1?The joint distribution of Y1…….,Yris an exponential family with density

[1/(2?)r][n!/(n-r)!]exp[-(?ri=1yi+(n-r)yr)/2?], 0<y1<=……..<=yr
and
Find the MLE of ?Y (maximum likelihood estimator of ?)
2?The distribution of [?ri=1Yi+(n-r)Yr]/? is ?2 with 2r degrees of freedom
3?Let Y1,Y2,……..denote the time required until the first, second,……event occurs in a Poisson process with parameter 1/2?’, Then Z1=Y1/?’, ?2=(?2-?1)/?’, ?3=(?3-?2)/?’,…….are independent distributed as ?2with 2 degrees of freedom,
and the joint density of Y1,……,Yris an exponential family with density
[1/(2?’)r]exp(-yr/2?’), 0<y1<=……<=yr .

The distribution of Yr/?’ is again ?2with 2r degrees of freedom.

4?The same model arises in the application to life testing if the number n of tubes is held constant by replacing each burned-out tube with a new one, and if Y1denotes the time at which the 1st tube burns out, Y2the time at which the 2nd tube burns out, and so on, measured from some fixed time. The lifetimes are assumed to be exponentially distributed.

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