Actuarial Outpost Hazard Rate and Exponential Dist.
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#1
10-17-2007, 11:52 AM
 actuarygal Member SOA Join Date: Aug 2007 Studying for FHR Module College: Xavier Alum Favorite beer: Wine please! Posts: 342
Hazard Rate and Exponential Dist.

Could someone explain what it means to say a random variable X has an exponential distribution with hazard rate 3? I can't seem to find a clear definition on this and it's really confusing me.
#2
10-17-2007, 11:59 AM
 Actuarialsuck Member Join Date: Sep 2007 Posts: 5,324

Quote:
 Originally Posted by actuarygal Could someone explain what it means to say a random variable X has an exponential distribution with hazard rate 3? I can't seem to find a clear definition on this and it's really confusing me.
Hazard rate being constant, I believe, implies exponential distribution (as far as cont. distributions go) and vice versa. So you are given
$h(x) \, = \, 3 \, = \, -\frac{S'(x)}{S(x)} \, = \, \frac{f(x)}{S(x)}$ etc. Does that help?
#3
10-17-2007, 02:18 PM
 actuarygal Member SOA Join Date: Aug 2007 Studying for FHR Module College: Xavier Alum Favorite beer: Wine please! Posts: 342

Let's break this down to what will help me on the test. If I get a problem that says X is an exponential with hazard rate 2 does this imply lamda = 2 or the mean =2?
#4
10-17-2007, 02:44 PM
 Noddy Member Join Date: May 2007 Posts: 2,179

The hazard (or failure) rate for a cont. r.v. is the ratio of the p.d.f. to the survival function. It is also -d/dx ln[Sx]

I believe its a constant in the case of the exponential distribution due to its memoryless property (ie. the expected time until failure is the same no mater where in the interval you start counting from).

for the exponential distribution the hazard rate looks to me like it is = to lambda = (1/mean) = 1/theta.

I referred to p. 90 Actex 2005 ed. and wikipedia.

Last edited by Noddy; 11-03-2007 at 08:59 PM..
#5
10-17-2007, 03:05 PM
 atomic Member CAS Join Date: Jul 2006 Posts: 4,088

Quote:
 Originally Posted by actuarygal Let's break this down to what will help me on the test. If I get a problem that says X is an exponential with hazard rate 2 does this imply lamda = 2 or the mean =2?
For an exponential distribution with density $\frac{1}{\theta}e^{-x/\theta}$, the survival function is $e^{-x/\theta}$. Hence the expected value is the integral of the survival function, which gives $\theta$. The hazard function is the density divided by survival, which is $h(x)=1/\theta$. Since this is a constant function of x, the hazard function is in this case the reciprocal of the mean. This makes intuitive sense because the larger the hazard function, the less likely the random variable will be observed to be larger.
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#6
10-17-2007, 03:27 PM
 actuarygal Member SOA Join Date: Aug 2007 Studying for FHR Module College: Xavier Alum Favorite beer: Wine please! Posts: 342

Got it. Thanks guys!
#7
10-19-2007, 12:07 AM
 Noddy Member Join Date: May 2007 Posts: 2,179

so actuarygals question could have simply read "a cont. r.v. has a hazard rate of three" and we know it must have an exponential distribution with a mean of 1/3, lambda=3, 1/theta = 3 (according to what actuarialsuck believes)

which means that the r.v. is exponential with a mean of 1/3 and that say for example the units are weeks, the mean is a third of a week between events and the hazard rate is 3 per week ?
#8
10-19-2007, 08:01 AM
 daaaave David Revelle Join Date: Feb 2006 Posts: 2,486

Quote:
 Originally Posted by Noddy which means that the r.v. is exponential with a mean of 1/3 and that say for example the units are weeks, the mean is a third of a week between events and the hazard rate is 3 per week ?
Yes. In this setting, you could also say the expected value of the number of events per week is 3.
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