Editing Actuarial notation (section)

===Life tables===
A [[life table]] (or a mortality table) is a mathematical construction that shows the number of people alive (based on the assumptions used to build the table) at a given age. In addition to the number of lives remaining at each age, a mortality table typically provides various probabilities associated with the development of these values.

<math>\,l_x</math> is the number of people alive, relative to an original cohort, at age <math>x</math>. As age increases the number of people alive decreases.

<math>\,l_0</math> is the starting point for <math>\,l_x</math>: the number of people alive at age 0. This is known as the '''radix''' of the table. Some mortality tables begin at an age greater than 0, in which case the radix is the number of people assumed to be alive at the youngest age in the table.

<math>\omega</math> is the limiting age of the mortality tables. <math>\,l_n</math> is zero for all <math>\,n \geq \omega</math>.

<math>\,d_x</math> is the number of people who die between age <math>x</math> and age <math>x + 1</math>. <math>\,d_x</math> may be calculated using the formula <math>\,d_x = l_x - l_{x+1}</math>

{| class="wikitable"
! style="text-align: center;" | <math>x</math>
! style="text-align: center;" | <math>l_x</math>
! style="text-align: center;" | <math>d_x</math>
|-
| style="text-align: center;" | 0
| style="text-align: center;" | <math>l_0</math>
| style="text-align: center;" | 
|-
| style="text-align: center;" | ...
| style="text-align: center;" | ...
| style="text-align: center;" | ...
|-
| style="text-align: center;" | <math>x</math>
| style="text-align: center;" | <math>l_x</math>
| style="text-align: center;" | <math>d_x=l_x-l_{x+1}</math>
|-
| style="text-align: center;" | <math>x+1</math>
| style="text-align: center;" | <math>l_{x+1}</math>
| style="text-align: center;" | <math>d_{x+1}</math>
|-
| style="text-align: center;" | ...
| style="text-align: center;" | ...
| style="text-align: center;" | ...
|-
| style="text-align: center;" | <math>\omega-1</math>
| style="text-align: center;" | <math>l_{\omega-1}</math>
| style="text-align: center;" | <math>d_{\omega-1}=l_{\omega-1}</math>
|-
| style="text-align: center;" | <math>\omega</math>
| style="text-align: center;" | 0
| style="text-align: center;" | 0
|}

<math>\,q_x</math> is the probability of death between the ages of <math>x</math> and age <math>x + 1</math>.

:<math>\,q_x = d_x / l_x</math>

<math>\,p_x</math> is the probability that a life age <math>x</math> will survive to age <math>x + 1</math>.

:<math>\,p_x = l_{x+1} / l_x</math>

Since the only possible alternatives from one age (<math>x</math>) to the next (<math>x+1</math>) are living and dying, the relationship between these two probabilities is:

:<math>\,p_x+q_x=1</math>

These symbols may also be extended to multiple years, by inserting the number of years at the bottom left of the basic symbol.

<math>\,_nd_x = d_x + d_{x+1} + \cdots + d_{x+n-1} = l_x - l_{x+n}</math> shows the number of people who die between age <math>x</math> and age <math>x + n</math>.

<math>\,_nq_x</math> is the probability of death between the ages of <math>x</math> and age <math>x + n</math>.

:<math>\,_nq_x = {}_nd_x / l_x</math>

<math>\,_np_x</math> is the probability that a life age <math>x</math> will survive to age <math>x + n</math>.

:<math>\,_np_x = l_{x+n} / l_x</math>

Another statistic that can be obtained from a life table is [[life expectancy]].

<math>\,e_x</math> is the curtate expectation of life for a person alive at age <math>x</math>. This is the expected number of complete years remaining to live (you may think of it as the expected number of birthdays that the person will celebrate).

:<math>\,e_x = \sum_{t=1}^{\infty} \ _tp_x</math>

A life table generally shows the number of people alive at integral ages. If we need information regarding a fraction of a year, we must make assumptions with respect to the table, if not already implied by a mathematical formula underlying the table. A common assumption is that of a Uniform Distribution of Deaths (UDD) at each year of age. Under this assumption, <math>\,l_{x+t}</math> is a [[linear interpolation]] between <math>\,l_x</math> and <math>\,l_{x+1}</math>. i.e.

:<math>\,l_{x+t} = (1 - t)l_x + tl_{x+1} </math>