Actuarial Outpost Equation 4.4 vs. Equation 4.5 in ASM
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#1
05-19-2009, 04:47 PM
 jvaldez84 Member Join Date: Feb 2007 Location: Houston, TX Studying for C College: Texas A&M Alumni Posts: 1,053
Equation 4.4 vs. Equation 4.5 in ASM

How do you know which one to use? I always get my integrands mixed up.

4.4) tpx = exp(-int(mu_s),x,x+t)
4.5) tpx = exp(-int(mu_x(s),0,t)
#2
05-19-2009, 05:06 PM
 billvp Member Join Date: Nov 2007 Posts: 303

use whichever you'd like
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#3
05-19-2009, 05:19 PM
 Bored Insomniac Member Join Date: May 2007 Location: Connecticut Posts: 83

The equations are exactly the same, but it is almost always easier to think in terms of 4.5.

Keep in mind that $\mu_{x+t}$ = $\mu_x(t)$. This means that the expression within the integrals, evaluated at the lower and upper bounds of the integrals, gives the same values. Take the equation for $\mu$ given in example 4A, part 1:

$\mu_{35}(t) = \mu_{35+t} = \frac1{100+t}$

In part 1, x=35 and t=10.

Using 4.4, you integrate $\mu_s$ from s = 35+0 to 35+10.

Using 4.5, s=t and you integrate $\mu_{35}(s)$ from s=0 to s=10. Due to the equality above, $\mu_{35}(0) = \mu_{35+0} = \mu_{35}$ and $\mu_{35}(10) = \mu_{35+10} = \mu_{45}$. Likewise they are equivalent at all values in between, and you are doing the exact same calculation in 4.4 and 4.5.
#4
05-19-2009, 05:52 PM
 jvaldez84 Member Join Date: Feb 2007 Location: Houston, TX Studying for C College: Texas A&M Alumni Posts: 1,053

In question 5.3: Calculate the probability that a 40-year-old will survive to age 42 if mortality follows Weibull's law: $\mu_{x}=kx^n$ with $k=\frac{1}{100}$ and $n=1$. Why do we integrate from 40 to 42 using 4.4 and not 0 to 2 using 4.5?
#5
05-19-2009, 06:27 PM
 Actuarialsuck Member Join Date: Sep 2007 Posts: 5,324

Because your hazzard rate is given in terms of a newborn, compare what you have to equation 4.4 there's a difference between mu_{x} and mu_{x+s} you have mu_{x}
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#6
05-19-2009, 09:31 PM
 CalBear07 Member SOA Join Date: Dec 2006 Location: Bay Area, CA Posts: 374

Generally, you use 4.4 if mu is given in terms of x, and 4.5 if mu is given in terms of t.
#7
05-20-2009, 09:47 AM
 jvaldez84 Member Join Date: Feb 2007 Location: Houston, TX Studying for C College: Texas A&M Alumni Posts: 1,053

Oh ok. I will try to approach problems using your suggestions. Thanks for the help.
#8
05-20-2009, 09:52 AM
 yanksrule08 Member Join Date: Aug 2007 Posts: 64

I think you are too wrapped up in the equations and trying to apply them without understanding the underlying principles. It's important to understand where these equations come from in order to do well on the exam.

This is how I think of these problems:

Starting at what age does the mortality rate apply? If it's just mu(x), then if I use the integral from 0 to 2 for a 40 year old living to 42, then I'm going to get the same probability for a newborn living 2 years, a 40 year old living 2 years, a 90 year old living 2 years, which isn't correct since the mortality function depends on x, the age of the person. So I need to be able to take into accoubnt the age of the person in my integral, i.e. an integral from x to x+s.

If the mortality rate were given as mu(40+t), then the mortality rate function is starting at the age 40. So if I wanted probability of a 40 year old living to 42, I would use an integral from 0 to 2. If I wanted probablity of a 50 year old living to 52, I would use an integral from 10 to 12.

Of course, a caveat to all of this is when mu(x) = a constant. In this case, mortality rate does NOT depend on the age of the person, in other words the probability you will die in any given span does not depend on how old you are (the distribution is memoryless). So in this case, probablity of a newborn living 2 years = probablity of a 40 year old living 2 year = probablity of a 300 year old living 2 years. So if you are given mu(x) = k, and want to find the probablity of a 40 year old living 2 years, you can use an integral from 0 to 2. Of course you could also use one form 40 to 42, but you would get the same answer.
#9
05-22-2009, 11:12 AM
 jvaldez84 Member Join Date: Feb 2007 Location: Houston, TX Studying for C College: Texas A&M Alumni Posts: 1,053

Quote:
 Originally Posted by yanksrule08 I think you are too wrapped up in the equations and trying to apply them without understanding the underlying principles. It's important to understand where these equations come from in order to do well on the exam. This is how I think of these problems: Starting at what age does the mortality rate apply? If it's just mu(x), then if I use the integral from 0 to 2 for a 40 year old living to 42, then I'm going to get the same probability for a newborn living 2 years, a 40 year old living 2 years, a 90 year old living 2 years, which isn't correct since the mortality function depends on x, the age of the person. So I need to be able to take into accoubnt the age of the person in my integral, i.e. an integral from x to x+s. If the mortality rate were given as mu(40+t), then the mortality rate function is starting at the age 40. So if I wanted probability of a 40 year old living to 42, I would use an integral from 0 to 2. If I wanted probablity of a 50 year old living to 52, I would use an integral from 10 to 12. Of course, a caveat to all of this is when mu(x) = a constant. In this case, mortality rate does NOT depend on the age of the person, in other words the probability you will die in any given span does not depend on how old you are (the distribution is memoryless). So in this case, probablity of a newborn living 2 years = probablity of a 40 year old living 2 year = probablity of a 300 year old living 2 years. So if you are given mu(x) = k, and want to find the probablity of a 40 year old living 2 years, you can use an integral from 0 to 2. Of course you could also use one form 40 to 42, but you would get the same answer.
Thanks for your explanation. I knew the formulas however I was not fully understanding like you had said. However, after reading this I believe I understand the difference between the two.

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