The length of time in hours that a rechargeable calculator battery will hold its charge is a random variable. Assume that this variable has a Weibull distribution with α=0.01 and β=2. (a) What is the density for X? (b) What are the mean and variance for X? (c) What is the reliability function for this random variable? (d) What is the reliability of such a battery at t=3 hours? At t=12 hours? At t=20 hours? (e) What is the Hazard rate function for these batteries?
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
The length of time in hours that a rechargeable calculator battery will hold its charge is a random variable. Assume that this variable has a Weibull distribution with α=0.01 and β=2.
(a) What is the density for X?
(b) What are the mean and variance for X?
(c) What is the reliability
(d) What is the reliability of such a battery at t=3 hours? At t=12 hours? At t=20 hours?
(e) What is the Hazard rate function for these batteries?
(f) What is the failure rate at t=3 hours? At t=12 hours? At t=20 hours?
(g) Is the hazard rate function an increasing or decreasing function? Does this seem to be reasonable from practical point of view? Explain
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