(8) Is the hazard rate function an increasing or a decreasing function? Does this seem to be reasonable from a practical point of view? Explain. 59 o Computer chips do not "wear out" in the ordinary sense. Assuming that de- fective chips have been removed from the market by factory inspection, it is 150 INTRODUCTION TO PROBABILITY AND STATISTICS reasonable to assume that these chips exhibit a constant hazard rate. Let sh. hazard rate be given by p(t) = .02. (Time is in years.) (a) In a practical sense, what are the main causes of failure of these chins? (b) What is the reliability function for chips of this type? bns nv (c) What is the reliability of a chip 20 years after it has been put into use? (d) What is the failure density for these chips? (e) What type of random variable is X, the time to failure of a chip? (f) What is the mean and variance for X? (g) What is the probability that a chip will be operable for at least 30 years? 60. The random variable X, the time to failure (in thousands of miles driven) of bozogang the signal lights on an automobile has a Weibull distribution with a = .04 and B = 2. (a) Find the density, mean, and variance for X. (b) Find the reliability function for X. (c) What is the reliability of these lights at 5000 miles? At 10,000 miles? (d) What is the hazard rate function? (e) What is the hazard rate at 5000 miles? At 10,000 miles? (f) What is the probability that the lights will fail during the first 3000 miles driven? tal1ls 61. Show that for a > 0 and B> 0, vib aßxB-le-ax dx = 1 %3D 01 thereby showing that the nonnegative function given in Definition 4.7.1 is a density for a continuous random variable. Hint: Let z = axº. 62. Let X be a Weibull random variable with parameters a and ß. Show that E[X²] = a¯2/ß[T(1 + 2/B). Hint: In evaluating %3D