14. Life testing. Let XI" ' " XII be independently distributed with exponential density (20) -le- x / 28 for x 0, and let the ordered X's be denoted by Y1 :::; Y2 :::; • • • :::; It is assumed that YI becomes available first, then Y2 , and so on, and that observation is continued until 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 a-particles have been emitted. (i) The joint distribution of l;, .. ., is an exponential family with density 1 n! l ty;+(n-r)Yr] __ ex _'=--"1 _ (20r (n - r)! p - 20 ' 0:::;YI:::; • .• s Yr ' (ii) The distribution of [L~ -I1'; + (n - r)~l/O is X2 with 2r degrees of freedom. (iii) Let YI , Y2 , . .. denote the time required until the first, second, . . . event occurs in a Poisson process with parameter 1/20' (see Chapter 1, Problem 1). Then ZI = YI/O', Z2 = (Y2 - YI)/O', Z3 = (lJ - Y2 )/0', ... are independently distributed as X2 with 2 degrees of freedom, and the joint density of YI , ... , is an exponential family with density 1 (Yr ) (20') r exp - 2iY ' 0:::;YI:::; ' " s Yr ' The distribution of ~/O' is again X2 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 YI 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 Z; = (n - i + 1)(1'; - 1';-1)/0 (i = 1,. .. , r) are independently distributed as X2 with 2 degrees of freedom, and [L~-I 1'; + (n - r)~l/O = L~_IZi ')