Suppose a and b are known and that 0 a b find the ml

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Unformatted text preview: equation, by integrating over the c axis: ∞ 0 (1 − FX (c))dc − E [X ] = FX (c)dc, (3.8) −∞ 0 or equivalently, by integrating over the u axis: 1 E [X ] = 0 − FX 1 (u)du. (3.9) The area rule can be justified as follows. As noted in Section 3.8.2, if U is a random variable uni− formly distributed on the interval [0, 1], then FX 1 (U ) has the same distribution as X. Therefore, it − −1 also has the same expectation: E [X ] = E [FX (U )]. Since FX 1 (U ) is a function of U, its expectation 1− − can be found by LOTUS. Therefore, E [X ] = E [FX 1 (U )] = 0 FX 1 (u)du, which proves (3.9), and hence, the area rule itself. 3.9 Failure rate functions Eventually a system or a component of a particular system will fail. Let T be a random variable that denotes the lifetime of this item. Suppose T is a positive random variable with pdf fT . The failure rate function, h = (h(t) : t ≥ 0), of T (and of the item itself) is defined by the following limit: P (t < T ≤ t + |T > t) h(t) = lim . →0 That is, given the item is still working after t time units, the probability the item fails withi...
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