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Unformatted text preview: , ' I'Ivii-EI‘TJI'IMMENTS “ﬂute we give examples of how p d f.s are created starting with a probabilltf
ﬁtkground. Let us ﬁrst consider the gamma distribution. Say we have a Poisson process (see comments at the end of Chapter 2) 1nvolv' - J
time intervals with mean A, and let T be the waiting time until the rth chan;.-.-'-
occurs With t > 0, the distribution function of T 15 given by k e—M F(t)=P(T5t)=1—P(T>t)=1—Z(———M)k!e since the latter summation is the probability of having less than r changes in t.--.
interval [0, I]. The derivative of F(t) is the p.d.f. of T, namely ' f(t) = Art"_le_)"/(r —1)!, 0 < t < 00. This is a gamma p.d.f. with 6 = 1/21 and a = r. So we can think of this p.d.f. as I - -.
appropriate model for a waiting time under certain assumptions.
Of course, if 01 = 1, then we obtain the exponential p.d.f. f(t) = (1/6)e_’/9, 0 <1 < 00.
It is interesting that the failure rate (hazard rate, force of mortality), _f_<x> (woe-W _ 1 1—F(,:——r—) 1—(1—e~l/9) _ 9’ is a constant. This means that if the length of life of people had an exponenti' .
distribution, the probability of a young man dying in the next year would
exactly the same as that of an old man. Hopefully 6 would be large so that Us:
would be small, and we would have found a mathematical “fountain of youthi'c'
Unfortunately, the failure rate usually increases for products and people.
Engineers often ﬁnd that the hazard (failure) rate is equal to af'l/ﬂ“, a :-
ﬂ > 0, which, by solving a simple differential equation, results in the Weibull p.dﬁ, xa— l a
f(x): eW’B) , 0 < x < oo. aﬂa A value of 01 around three frequently provides a good model fur the life of man}
engineering products. On the other hand, actuaries ﬁnd that the force of morta
of humans is exponential, like ae’”, a > 0, b > 0, which gives the Gompertz p.d.f‘E f(x) = abbre—(l/bxw‘l), 0 < x < oo. Chapter Three Comments 155 For illustration, they say that the force of mortality of males often increases about
10 percent per year, meaning the probability of a male dying within one year
doubles about every 72/10 = 7.2 years, by the "Rule 01H." Thank goodness the
force of mortality is extremely low for a ﬁve-year old boy, possibly about 6.00004:
that is, the probability that such a boy will die in the next year is about 4 out of
100,000. This, however, is much higher than the probability winning “power ball." ...
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- Fall '07