Let be the standard normal density function and let be the standard normal cumulative distribution function. By using a taylor series expansion of around 0, show that:𝐹(𝑥) , where𝐹(𝑥) = 1

(a) Let be the standard normal density function and
let be the standard normal cumulative distribution
function.
By using a taylor series expansion of around 0, show that:𝐹(𝑥)
, where𝐹(𝑥) = 1
2 + 𝐺(𝑥)
61K_ZN6c
(b) By using MS Excel, and the result from (a) approximate the
CDF for a life aged 0 with future lifetime random
variable . Terminate your Taylor series at n=3 and𝑍 ∼ 𝑁(0, 1)
include the following fields in your excel sheet:
Age: this should range from 0 to 1.8. Increase in increments of
0.1.
Approx Normal Cumulative Probabilities
(c) Adjust the spreadsheet from (b) to approximate the survival

function for a life with survival function in assumption 1.
Include the following fields:
Adjusted Age: use the transformation , where Z is a𝑍 = 𝑋−μ
σ
standard normal random variable, and are mean and varianceμ σ
of X respectively where .𝑋 ∼ 𝑁(75, 1)
Adjusted Approx Normal Cumulative Probabilities: Hint: to
Survival function: use Adjusted Approx Normal Cumulative
Probabilities to calculate.
make the adjustment compute .𝐹𝑥(𝑠) = 𝑃(𝑋 < 𝑥 + 𝑠|𝑋 > 75)
(d) Consider (2) lives aged 75 with future lifetimes which following
Mortality assumptions:
𝑠𝑝𝑥 = 𝑃(𝑋 > 𝑥 + 𝑠|𝑋 > 75)
𝑠𝑝𝑦 = 𝑃(𝑌 > 𝑦 + 𝑠|𝑌 > 75)
where and assume both and𝑋 ∼ 𝑁(75, 1) 𝑌 ∼ 𝑁(75, 1) 𝑋 𝑌
are independent.Using the results from (c) Calculate:
and (survival probabilities for 1 year for both the𝑝75:75 𝑝75:75
joint and last survivor status respectively).
Total: 20 marks

Question 2:
Consider the following ASD (Associated Single Decrement) survival
functions for (x):
, t > 0𝑡𝑝’(1)
𝑥 = 𝑒−0.05𝑡
= Uniform distribution on the interval [0,45]𝑡𝑝’(2)
𝑥
= , t > 0𝑡𝑝’(3)
𝑥 𝑒−0.10𝑡
Where (1), (2), (3) represent Dismissal, withdrawal and retirement
respectively. Determine the following:
1. .𝑡𝑝 (τ)
𝑥
2. 𝑡𝑞(τ)
𝑥 .
3. If A and B are independent show that and are𝐴‘ 𝐵‘
independent (i.e. their complements are independent).
4. Hence, Prove the following relationship:
= +( +𝑡𝑞(𝑖)
𝑥 𝑡𝑞‘(𝑖)
𝑥𝑡𝑝‘(𝑗)
𝑥𝑡𝑝‘(𝑘)
𝑥 𝑡𝑞‘(𝑖)
𝑥𝑡𝑞‘(𝑗)
𝑥𝑡𝑝‘(𝑘)
𝑥 𝑡𝑞‘(𝑖)
𝑥𝑡𝑝‘(𝑗)
𝑥
) +𝑡𝑞‘(𝑘)
𝑥 𝑡𝑞‘(𝑖)
𝑥𝑡𝑞‘(𝑗)
𝑥𝑡𝑞‘(𝑘)
𝑥

Where i,j and k are 3 distinct modes of decrement.
Hint: ASD assumes independence of for survival for each
mode of Associated Single Decrement to construct multiple
decrements. Also, consider 3 cases.
5. Use the relationship in part (4) to calculate 2 𝑝(3)
𝑥
Total: 15 marks