Suppose that n electronic components, each having an exponentially distributed length of
Chapter 6, Problem 6.91(choose chapter or problem)
Suppose that n electronic components, each having an exponentially distributed length of life with mean , are put into operation at the same time. The components operate independently and are observed until r have failed (r n). Let Wj denote the length of time until the jth failure, with W1 W2 Wr. Let Tj = Wj Wj1 for j 2 and T1 = W1. Notice that Tj measures the time elapsed between successive failures. a Show that Tj , for j = 1, 2,...,r, has an exponential distribution with mean /(n j +1). b Show that Ur = r j=1 Wj + (n r)Wr = r j=1 (n j + 1)Tj and hence that E(Ur) = r. [Ur is called the total observed life, and we can use Ur /r as an approximation to (or estimator of ) .]
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