Age x last birthday 1 June > Age x last birthday 30 June

if **P _{x,t}** is the population aged x last birthday on 1

^{st}January,

and **P ^{*}_{x,t}** is the population aged x last birthday on 30

^{th}June,

give an approximation for **Px,t** in terms of **P*x,t**

if P_{x,t} is the population aged x last birthday on 1^{st} January,

and P^{*}_{x,t} is the population aged x last birthday on 30^{th} June,

**P _{x,t} = ^{1}/_{2}(P^{*}_{x-1,t} + P^{*}_{x,t})**

Age x nearest birthday 1st Jan > Age x last birthday 1st Jan

if **P _{x,t}** is the population aged x last birthday on 1

^{st}January,

and **P ^{*}_{x,t}** is the population aged x nearest birthday on 1st January,

give an approximation for **Px,t** in terms of **P*x,t**

if P_{x,t} is the population aged x last birthday on 1^{st} January,

and P^{*}_{x,t} is the population aged x nearest birthday on 1^{st} January,

**P _{x,t} = ^{1}/_{2}(P^{*}_{x,t} + P^{*}_{x+1,t})**

Calendar Year of Death - Calendar Year of Birth > Exposed to Risk for μ_{x+f }

The deaths during the period of investigation, θ_{x} , have been classified by age x at the date of death, where

*x = calendar year of death - calendar year of birth.*

Derive an expression for the exposed to risk in terms of the P_{x}(t) which corresponds to the deaths data and which may be used to estimate the force of mortality, μ_{x+f} at age x + f.

A census of those aged x next birthday on 1 January in each year would correspond to the classification of deaths.

But we have lives classified by age x last birthday.

However, the number alive aged x next birthday on any date is equal to the number alive aged x - 1 last birthday.

The number alive aged x - 1 last birthday on 1 January in year t is given by P_{x-1}(t).

At the end of year t this cohort will be aged x last birthday.

Thus, using the trapezium rule, the correct exposed to risk at age x in year t is given by

^{1}/_{2}[P_{x-1}(t) + P_{x}(t+1)]

Assuming birthdays are uniformly distributed over the calendar year, the average age at the start of the rate interval will be x - ½.

Therefore the average age in the middle of the rate interval is x.

Assuming a constant force of mortality between x - ½ and x + ½, therefore,

f = 0.

Example Exam Q References

March 2006 - B4 (p 69)

September 2006 - B2 (p. 97)

April 2007 - Q4 (p. 125)

Sep 2007 - Q8 (p. 157)

Apri 2008 - Q5 (p. 180)

Sept 2008 - Q6 (p. 209)

April 2009 - Q 6 (p. 236)

Sept 2009 - Q 4 (p. 262)

Apr 2010 - Q6 (p. 288)

Sep 2010 - Q 7 (p. 321)

Sep 2011 - Q 9 (p. 382)

Sep 2012 - Q 3 (p. 433)