3.
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!
A system consists of five identical components connected in series as shown:
As soon as one component fails, the entire system will fail. Suppose each component has a lifetime that is exponentially distributed with λ = .01 and that components fail
independently of one another. Define events Ai ={ith component lasts at least t hours}, i =1,…, 5, so that the Ais are independent events. Let X = the time at which the system fails— that is, the shortest (minimum) lifetime among the five components
a. The
,…, A5?
b. Using the independence of the Ai ′s, compute P(X ≥ t). Then obtain F(t) = P(X ≥ t) andthe
c. Suppose there are n components, each having exponentiallifetime with parameter l.What type of distributiondoes X have?
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