The lifetime of a certain computer component (in hours) is well modeled with a Weibull distribution with shape parameter β = 3 and scale parameter δ = 5000. Let X denote the lifetime of the component. Determine the following: (a) The probability that the component lasts more than 4000 hours, that is P (X > 4000). (b) The probability that the component lasts in total more than 8000 hours, given that it already lasted 4000 hours. In other words, P (X > 8000|X > 4000). (c) Compare your answers to “a” and “b” and comment on their relation. (d) Repeat questions “a” to “c” for the case when one has β = 0.5. Note: the shape parameter β can be useful to model different situations. If β > 1 this distribution is adequate to model parts that are well manufactured, but suffer wear and tear with normal use. On the other hand, if β < 1 this distribution can be used to model the lifetime of low quality parts that either fail almost immediately, or else can last a long time.
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 lifetime of a certain computer component (in hours) is well modeled with a Weibull distribution with shape parameter β = 3 and scale parameter δ = 5000. Let X denote the lifetime of the component. Determine the following:
(a) The probability that the component lasts more than 4000 hours, that is P (X > 4000).
(b) The probability that the component lasts in total more than 8000 hours, given that it already lasted 4000 hours. In other words, P (X > 8000|X > 4000).
(c) Compare your answers to “a” and “b” and comment on their relation.
(d) Repeat questions “a” to “c” for the case when one has β = 0.5.
Note: the shape parameter β can be useful to model different situations.
If β > 1 this distribution is adequate to model parts that are well manufactured, but suffer wear and tear with normal use. On the other hand,
if β < 1 this distribution can be used to model the lifetime of low quality parts that either fail almost immediately, or else can last a long time.
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