Name(s):                                                            :
Order Statistics Break the Ice Problem
Tom Linton, February 23, 2000
  1. A light fixture in Tom's study has 4 light-bulbs. Tom can comfortably read in the study so long as two of the bulbs are not burned-out. Each bulb (independent of the others) has a lifetime of X hours, where X is an exponential random variable with an expected value of 1000 hours (so f(x) = 0.001*e^(-x / 1000), for x > 0).
    1. Find a formula for the cumulative distribution function for X, namely F(x) = P[X < x].














    2. The lifetimes of the 4 bulbs can be referred to as X1, X2, X3 and X4 (in no special order, so X1 is NOT the first to burn out etc.), and the order statistics of X1 to X4 can be denoted by Y1 (the smallest), Y2 (the second smallest), Y3 (the third smallest) and Y4 (the largest). Let gi(t) denote the density function for Yi, and Gi(t) denote the cumulative distribution function corresponding to gi(t) or Yi. Assume that Tom never replaces any bulbs that burn out. Express in common language what 1 - G4(2000) represents.







    3. Find a formula for g2(t).





    4. Notice that Tom can read in his study until the third bulb burns out. How likely is it that this time exceeds t = 1500 hours? This is probably easiest to calculate as 1 - G3(1500).