Then e x is the area of the region in the strip to

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Unformatted text preview: BLE 107 [0, c] ∪ [1, 1 + c] ∪ [2, 2 + c] ∪ · · · , indicated in bold in the figure. Therefore, FR (c) = P {X − X ≤ c} ∞ P {k ≤ X ≤ k + c} = k=0 ∞ k+c = k=0 ∞ ∞ λe−λu du = k k=0 e−λk (1 − e−λc ) = = k=0 where we used the formula 1 + α + α2 + · · · = Differentiating FR yields the pmf: fR (c) = e−λk − e−λ(k+c) 1 1−α for the sum of a geometric series, with α = e−λ . λe−λc 1−e−λ 0 1 − e−λc , 1 − e−λ 0≤c≤1 otherwise. To find the limit of the pdf of R as λ → 0, apply l’Hospital’s rule to get lim fR (c) = λ→0 1 0≤c≤1 0 otherwise. The limit of fR is the pdf for the uniform distribution on the interval [0, 1]. Intuitively, the remainder R is nearly uniformly distributed over [0, 1] for small λ because for such λ the density of X is spread out over a large range of integers. Example 3.8.10 This example illustrates many possible particular cases. Suppose X has a pdf fX which is supported on an interval [a, b]. Suppose Y = g (X ) where g is a strictly increasi...
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