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Unformatted text preview: n the next time units is h(t) + o( ). 112 CHAPTER 3. CONTINUOUS-TYPE RANDOM VARIABLES The failure rate function is determined by the distribution of T as follows: P {t < T ≤ t + } P {T > t} FT (t + ) − FT (t) = lim →0 (1 − FT (t)) fT (t) = , 1 − FT (t) h(t) = lim →0 (3.10) because the pdf fT is the derivative of the CDF FT . ∞ Conversely, a nonnegative function h = (h(t) : t ≥ 0) with 0 h(t)dt = ∞ determines a probability distribution with failure rate function h as follows. The CDF is given by F (t) = 1 − e− Rt 0 h(s)ds . (3.11) It is easy to check that F given by (3.11) has failure rate function h. To derive (3.11), and hence show it gives the unique distribution with failure rate function h, start with the fact that we would t like F /(1 − F ) = h. Equivalently, (ln(1 − F )) = −h or ln(1 − F ) = ln(1 − F (0)) − 0 h(s)ds, which is equivalent to (3.11). Example 3.9.1 (a) Find the failure rate function for an exponentially distributed random variable with parameter λ. (b) Find the distribution with the linear failure rat...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.

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