Problem1-11 - STA 6934 Problem 1.11 (Solution prepared by...

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Unformatted text preview: STA 6934 Problem 1.11 (Solution prepared by Vladimir L. Boginski) Weibull distribution We(α , β ) : The density is given by: f (x α , β ) = for x ∈ 4 + , α > 0 , β > 0 . βæxö α çα ÷ èø β −1 e æ xö −ç ÷ èα ø β , The cdf can be expressed as follows: β F ( x) = ò f (t α , β )dt = ò α 0 0 x = −e ætö −ç ÷ èα ø β x x = 1− e æ xö −ç ÷ èα ø ætö ç÷ èα ø β −1 e ætö −ç ÷ èα ø β x dt = ò e ætö −ç ÷ èα ø 0 β æ æ t öβ ö d çç ÷ ÷ = çèα ø ÷ è ø β . 0 So we see that the cdf of the Weibull distribution can be written explicitly. The hazard rate: f (t ) h(t ) = = 1 − F (t ) β α ætö ç÷ èα ø β −1 1 − (1 − e e ætö −ç ÷ èα ø ætö −ç ÷ èα ø β ) β βætö =ç÷ α èα ø β −1 . It can be seen that the scale parameter α determines the behavior of the hazard rate: the function h(t) grows faster (w.r.t. t) for smaller values of α, and it grows slower for bigger values of α. ...
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This note was uploaded on 01/15/2012 for the course STA 6934 taught by Professor Young during the Fall '08 term at University of Florida.

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