Life testing. Let X1,...,Xn be independently distributed with exponential density (2) 1ex/2 for x 0, and

Life testing. Let X1,...,Xn be independently distributed with exponential density (2θ)

−1e−x/2θ for x ≥ 0, and let the ordered X’s be denoted by Y1 ≤ Y2 ≤ ··· ≤ Yn. It is assumed that Y1 becomes available first, then Y2, and so on, and that observation is continued until Yr has been observed. This might arise, for example, in life testing where each X measures the length of life of, say, an 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 α-particles have been emitted.

(i) The joint distribution of Y1,...,Yr is an exponential family with density 1

(2θ)r n!

(n − r)! exp

⎡

⎢

⎢

⎣−

r i=1 yi + (n − r)yr 2θ

⎤

⎥

⎥

⎦ , 0 ≤ y1 ≤···≤ yr.

(ii) The distribution of [r i=1 Yi+(n−r)Yr]/θ is χ2 with 2r degrees of freedom.

(iii) Let Y1, Y2,... denote the time required until the first, second, . . . event occurs in a Poisson process with parameter 1/2θ
(see Problem 1.1). Then Z1 = Y1/θ

, Z2 = (Y2 − Y1)/θ

, Z3 = (Y3 − Y2)/θ

,... are independently distributed as χ2 with 2 degrees of freedom, and the joint density Y1,...,Yr is an exponential family with density 1

(2θ
)r exp 

− yr 2θ



, 0 ≤ y1 ≤···≤ yr.

The distribution of Yr/θ
is again χ2 with 2r degrees of freedom.
(iv) 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 Y1 denotes the time at which the first tube burns out, Y2 the time at which the second tube burns out, and so on, measured from some fixed time.
[(ii): The random variables Zi = (n − i + 1)(Yi − Yi−1)/θ (i = 1, 2,...,r) are independently distributed as χ2 with 2 degrees of freedom, and [r i=1 Yi + (n −
r)Yr/θ = r i=1 Zi.]