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Ark raw
Member
- why is the force of mortality μ_x defined as, μ_x =-d(lnl_x)/dx ?
Thank You.
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Thank You.
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Robert Chadburn
Member
For your first question , have you read Section 1.3 of Chapter 3? - it is fully explained there. Let me know if this does not explain it for you.
For your second question I'll repeat here I what put for Bharti in an earlier thread that he started a little while ago, where he asked more or less the same question. (Note it's always worth having a quick scan of recent threads to see if your query is already covered.) The numbers below are taken from the AM92 ultimate table, in the Orange Tables book. Here is my explanation:
mu(x) is the annual rate at which a person is dying at the exact age of x (so if x is an integer, we mean on the exact date at which the person has their x birthday).
This rate is changing continuously as age changes continuously. So, (Tables p81), anyone alive at exact age 100 will be dying at an annual rate of 0.421777. Anyone alive at age 101 will be dying at an annual rate of 0.457202. Someone aged 100.5 would be dying at an annual rate somewhere between the two.
To illustrate the idea of a rate, let's assume the rate (mu) is constant over the year of age (100 to 101). If we do this then we should use the formula on page 14 of Chapter 3 to calculate this "average" mu. So we calculate mu = -ln(p100) which comes to 0.44 (which is somewhere between the actual rates at age 100 and 101, as we'd expect).
Now let's put 1000 people all aged between 100 and 101 in a house. If someone dies (or they become 101) they have to leave the house immediately and are immediately replaced by another person who is then aged between 100 and 101. So, this means there are always 1000 people in this house all aged between 100 and 101 all the time forever. Suppose we count how many people die in the house during a year? We would expect that to be 1000 x 0.44 = 440. The rate of dying would be forever 440 per year as there will always be 1000 people in the house and they will all be subject to the mortality rate of 0.44 pa per person.
Now to compare this with qx. So, now let's start with an empty house and put 1000 people, all aged exactly 100, into the house at the same time. No-one is replaced when they die, the house just gets emptier! Now we just count how many of these people die during the coming year. As before they will all be subject to the same rate of mortality during the year of age (as we have assumed it is constant), but as the number of survivors falls during the year the total deaths we'd expect to see would be fewer than 440. The expected number of deaths in this case would be 1000 x q100, as q100 is the probability of a person aged 100 dying before reaching age 101. So, the expected number of deaths would only be 1000 x 0.3555 = 356.
Robert
For your second question I'll repeat here I what put for Bharti in an earlier thread that he started a little while ago, where he asked more or less the same question. (Note it's always worth having a quick scan of recent threads to see if your query is already covered.) The numbers below are taken from the AM92 ultimate table, in the Orange Tables book. Here is my explanation:
mu(x) is the annual rate at which a person is dying at the exact age of x (so if x is an integer, we mean on the exact date at which the person has their x birthday).
This rate is changing continuously as age changes continuously. So, (Tables p81), anyone alive at exact age 100 will be dying at an annual rate of 0.421777. Anyone alive at age 101 will be dying at an annual rate of 0.457202. Someone aged 100.5 would be dying at an annual rate somewhere between the two.
To illustrate the idea of a rate, let's assume the rate (mu) is constant over the year of age (100 to 101). If we do this then we should use the formula on page 14 of Chapter 3 to calculate this "average" mu. So we calculate mu = -ln(p100) which comes to 0.44 (which is somewhere between the actual rates at age 100 and 101, as we'd expect).
Now let's put 1000 people all aged between 100 and 101 in a house. If someone dies (or they become 101) they have to leave the house immediately and are immediately replaced by another person who is then aged between 100 and 101. So, this means there are always 1000 people in this house all aged between 100 and 101 all the time forever. Suppose we count how many people die in the house during a year? We would expect that to be 1000 x 0.44 = 440. The rate of dying would be forever 440 per year as there will always be 1000 people in the house and they will all be subject to the mortality rate of 0.44 pa per person.
Now to compare this with qx. So, now let's start with an empty house and put 1000 people, all aged exactly 100, into the house at the same time. No-one is replaced when they die, the house just gets emptier! Now we just count how many of these people die during the coming year. As before they will all be subject to the same rate of mortality during the year of age (as we have assumed it is constant), but as the number of survivors falls during the year the total deaths we'd expect to see would be fewer than 440. The expected number of deaths in this case would be 1000 x q100, as q100 is the probability of a person aged 100 dying before reaching age 101. So, the expected number of deaths would only be 1000 x 0.3555 = 356.
Robert
A
Ark raw
Member
I've read that section it doesn't explain the reason behind this definition.
Secondly, thank you for explaining that 2nd answer.(sorry, I forgot to give a quick scan for a similar thread, will keep this in my mind from here on)
Thank you
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Robert Chadburn
Member
Please tell me what it is in this section that you don't understand, as it starts with the definition of mu and ends with the formula that you quote! I presume you are happy with at least some of the steps?
Thanks
Robert
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Ark raw
Member
the very 1st definition of mu_x i.e.
mu_x= lim (1/h)*P[T=<x+h|T>x]
why is mu_x defined this way ?
mu_x= lim (1/h)*P[T=<x+h|T>x]
why is mu_x defined this way ?
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Robert Chadburn
Member
Thank you for clarifying your question 
mu_x is the annual rate at which a person aged x is dying at exact age x.
P[T=<x+h|T>x] is the probability that a life who is aged x dies in the next h of a year.
Suppose h was 1 day (= 1/365 of a year). And say the probability of dying in this period was 0.00004. If we divide this by h, we get the annual rate at which the person is dying over that single day. So this person is dying at an annual rate of 0.00004/(1/365) = 0.0146 pa over this single day.
By taking h to be a smaller and smaller time period we end up with the annual rate at which the person is dying at the current instant of time (ie at exact age x) - which is what mu_x is.
As here it very helpful if you make your question as clear as possible - this will help others to answer your queries as well as me - particularly as you are asking quite a lot of questions!
Thank you!
Robert
mu_x is the annual rate at which a person aged x is dying at exact age x.
P[T=<x+h|T>x] is the probability that a life who is aged x dies in the next h of a year.
Suppose h was 1 day (= 1/365 of a year). And say the probability of dying in this period was 0.00004. If we divide this by h, we get the annual rate at which the person is dying over that single day. So this person is dying at an annual rate of 0.00004/(1/365) = 0.0146 pa over this single day.
By taking h to be a smaller and smaller time period we end up with the annual rate at which the person is dying at the current instant of time (ie at exact age x) - which is what mu_x is.
As here it very helpful if you make your question as clear as possible - this will help others to answer your queries as well as me - particularly as you are asking quite a lot of questions!
Thank you!
Robert
A
Ark raw
Member
D
Darshan Mody
Member
Thank you for clarifying your question
mu_x is the annual rate at which a person aged x is dying at exact age x.
P[T=<x+h|T>x] is the probability that a life who is aged x dies in the next h of a year.
Suppose h was 1 day (= 1/365 of a year). And say the probability of dying in this period was 0.00004. If we divide this by h, we get the annual rate at which the person is dying over that single day. So this person is dying at an annual rate of 0.00004/(1/365) = 0.0146 pa over this single day.
By taking h to be a smaller and smaller time period we end up with the annual rate at which the person is dying at the current instant of time (ie at exact age x) - which is what mu_x is.
As here it very helpful if you make your question as clear as possible - this will help others to answer your queries as well as me - particularly as you are asking quite a lot of questions!
Thank you!
Robert
Hi
This has really been helpful but just one more query:
"the annual rate at which the person is dying at the current instant of time (ie at exact age x)"
this is the definition of mu(x) as you said but how do we interpret this in real life?
As in what does it mean when we say annual rate of dying BUT at the current instant of time?
"this person is dying at an annual rate of 0.00004/(1/365) = 0.0146 pa over this single day.
Like how exactly does this make sense? because we have an annual rate over a single day.
Thanks a ton!