Continuous type random variables the tildes on the

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Unformatted text preview: e parameter λ. Then P {T > t} = e−λt . It follows that P (T > s + t|T > s) = = P {T > s + t, T > s} P {T > s } P {T > s + t} P {T > s } e−λ(s+t) e−λs −λt =e = P {T > t}. = That is, P (T > s + t|T > s) = P {T > t}. This is called the memoryless property for continuous time. If T is the lifetime of a component installed in a system at time zero, the memoryless property of T has the following interpretation: Given that the component is still working after s time units, the probability it will continue to be working after t additional time units, is the same as the probability a new component would still be working after t time units. As discussed in Section 3.9, the memoryless property is equivalent to the failure rate being constant. Connection between exponential and geometric distributions As just noted, the exponential distribution has the memoryless property in continuous time. Recall from Section 2.5 that the geometric distribution has the memoryless property in discrete time. We shall further illustrate the close connection between the exponential an...
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