Life Tables:

It is a computational tool based on a specific survival model. Our
task is to generate a survival model and our output will be a life table. Life
table is based on Unitary method.

l_{x}^{ }= Expected no. of lives at age x

d_{x}= Expected no. of deaths between exact age x and exact
age x+1

NOTE: We call it expected and not actual because most of the values
will be in decimal too, so how can it be actual.

__But
there is a limitation of Life Tables: __It is defined only for
integer ages. If we have to calculate probability of death/survival at any
non-integer age we will use 3 assumptions:

1.**CFM= **Constant Force of
Mortality

2.**UDD= **Uniform
Distribution of death

3.**Balducci assumption**

1.CFM

· It
assumes that the force of mortality is constant, i.e. u_{x+t }= u_{x}

· If the
force of mortality of a newborn is constant, it means that the expected future
lifetime of this life is 1/μ no matter what age he is in.

· Survival
Function in this case will decrease exponentially.

· If
0<s<t<1 then _{t-s}p^{x+s} = _{t}p^{x}/_{s}p^{x}
= (p_{x})^{t-s}

^{}

^{}

2.UDD

· It assumes
that force of mortality increases between integer ages and here _{t}q^{x}
= t.q_{x} where 0<t<1

· It
assumes that deaths are uniform over a period of time. For ex: There are 24
deaths in a year then it means there are 2 deaths in a month or 1 death in every
15 days.

· It
assumes that PDF of T_{x} to be constant if it follows uniform distribution.

· Survival
Function in this case will decrease linearly.

· If 0<s<t<1 then _{t-s}q^{x+s}
= [q_{x}(t-s)]/[1-s.q_{x}]

3.Balducci:

· Here
force of mortality decreases between integer ages and assumption is _{1-t}q^{x+t}
= (1-t)q_{x}

· It means
that mortality is heaver in the first half of the year than it is in the second
half.

· The force
of mortality decreases in late teenage years/early twenties, so the Balducci
assumption may be more appropriate there. Look at the values for mu in the AM92
tables.

· If 0<s<t<1
then _{t-s}q^{x+s} = [q_{x}(t-s)]/[1-(1-t)q_{x}]

Now let’s see the human mortality graph:

· You can
see here the hump between ages 15 -20 years of age. This effect is called the
"accident hump", which is described in CT5. Basically, mortality
comes to a small peak around 17/18/19, and then decreases. The peak is usually
associated with deaths from car accidents of inexperienced drivers, and
drug/alcohol-related deaths.

But
there is something more related to mortality:

__Gompertz
and Makeham Laws of Mortality:__

Gompertz
law is an exponential function and it is often a reasonable assumption for
middle ages and older ages. Formula is u_{x} = Bc^{x}. Now we
say that it is a reasonable assumption between 35-65 years of age because if we
take the log of the equation i.e logu_{x}= logB + xlogc , then we can see
that increase in rate of mortality with age is constant which is often a
reasonable assumption.

Makeham
law formula is u_{x}= A+Bc^{x}. It incorporates a constant
term, which is sometimes interpreted as an allowance for accidental detahs, not
depending on age.

Gompertz
and makeham family curves equation is (polynomial{1})+exp(polynomial{2})

__Initial
and central rates of mortality__

q_{x}
= is called initial rates of mortality, because it is the probability that a
life alive at exact age x dies before the exact age x+1

m_{x}
= is the probability of dying between exact ages x and x+1 per person year lived
between exact ages x and x+1

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