If T is a continuous random variable that is always positive...

If T is a continuous random variable that is always positive (such as waiting time, lifetime, etc.), then the hazard function (called also force of mortality, or failure rate) is defined by h(t) = lim_{Delta o0} P[t le T < t+Delta mid T ge t] / Delta. The above probability expresses the probability of failure of an item, in the time interval [t, t+Delta t] given that it has survived up to time t. [4] a. Show that h(t) = f(t)/(1 - F(t)), where f and F are the p.d.f and the c.d.f of T, respectively. [4] b. We say that T has a Weibull distribution with parameters alpha, eta if the p.d.f is f(t) = { alphaeta^alpha t^{alpha-1} e^{-(eta t)^alpha}, t > 0, 0, t le 0. } The c.d.f. of a Weibull r.v. having parameters alpha and eta is given by F(t) = { 1 - e^{-(eta t)^alpha}, t > 0, 0, t le 0. } Show that for such a distribution, the hazard function h(t) is increasing for alpha > 1 and decreasing for alpha < 1. [4] c. Show that h(t) is equal to a constant if T has exponential distribution with parameter lambda. Find this constant.

Transcript

00:01 Ok, given a continuous random variable t, t is continuous, we define ht is some function which we define to be the limit when delta goes to 0.

00:16 The probability of t is greater or equal to little t and less or equal to little t plus delta t, delta given capital t is greater or equal to little t over delta.

00:30 Okay, then we want to show, first we want to show ht is equal to ft over 1 minus capital ft, where f is the pdf and capital is the cdf for this random variable t, capital t.

01:01 Okay, now suppose this conditional expectation, this conditional probability is meaningful.

01:10 That means the probability of, let's suppose this is greater than zero.

01:21 Why do we need this assumption? because if this probability is equal to zero, then our conditional probability is meaningless.

01:31 Okay, i mean this conditional probability is only defined for the condition, for the probability of the condition is non -zero.

01:44 Wrong.

01:47 We only need to do this proof under this assumption.

01:55 Once we have this assumption, we can use the formula for the conditional probability, i mean probability of delta t less or equal to t.

02:06 This conditional probability can be written as this event and t greater or equal to little t over the probability of capital t greater or equal to t.

02:24 Okay and it's very easy for us to see the first event is contained in the second one so if we take the union, take the intersection of them this term will disappear that means this conditional probability is equal to the probability of t greater or equal to little t and less or equal to little t plus delta t over the probability of capital t greater or equal to t.

02:51 Okay by the property of the the cdf random variable this is just equal to bar minus ft.

03:01 Okay, so to prove this identity, we only need to prove the limit when delta goes to zero, the probability of little t less or equal to capital t less or equal to t plus delta over delta is equal to f.

03:24 This is easy because as t is a continuous random variable, we know there is a density function for t, that means this probability can be written as some integral for the density function integral from t to t plus delta f dx over delta.

03:47 Okay, then use the fundamental theory of calculus, we know this guy is equal, it's just equal to f.

03:54 Why? because use the mean value theory for the integral, we know this guy, this integral, can be written as delta, which is t plus delta minus t, times f.

04:08 Okay, i mean this thing is just becoming just like that.

04:14 So 2 delta will be q, and c can be written as something between t and t plus delta, so it can be written as t plus c times delta, where c is greater or equal to 0 and less or equal to 1.

04:29 So this limit is equal to the limit when delta goes to zero at t plus theta times delta.

04:42 When delta goes to zero, again by the property of the density function, we know this.

04:54 In fact, to make our definition more rigorous, this should be the limit from the right.

05:02 I mean, delta goes to zero from the right.

05:06 Otherwise you can see this probability.

05:09 When delta is less or equal to zero, then this probability is equal to zero.

05:15 So our definition can be can be meaningless.

05:23 Okay, so here again this should be the right limit, the right limit, the right limit.

05:32 Okay, then for any density function, we know it must be continuous from the right.

05:39 So take the limit from the right.

05:42 Okay this is just equal to f.

05:52 Now we want to consider another question.

05:56 We are given the weibull distribution.

06:05 Weibull distribution with parameter alpha and beta.

06:12 That means our density is equal to alpha times beta to the power alpha times t to the power alpha minus 1 times e to the power negative beta times t to the power alpha.

06:31 For t greater than 0, it is equal to 0 else...

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