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(π‘₯) , where𝐹(π‘₯) = 1

Actuarial Mathematics 2: Assignment 2
Instructions:
● Assignment consists of 2 questions. Attempt all questions.
● Questions are to be compiled into one document.
● Questions can be written or typed.
● Written questions can be scanned and compiled into a single pdf
using the following application:
https://play.google.com/store/apps/details?id=com.intsig.camscanner
https://apps.apple.com/us/app/camscanner-pdf-scanner-app/id38862
7783
● Assignment should be submitted via googledrive by 12 midnight
Monday, November 21,2022.
Question 1:
We are analyzing the mortality of individuals after the age of 75 years.
We assume the following:
.𝑠𝑝π‘₯ = 𝑃(𝑋 > π‘₯ + 𝑠|𝑋 > 75)
i.e. X follows a normal distribution with mean𝑋 ∼ 𝑁(75, 1)
75 and variance 1.
, where S is the future lifetime random𝑆 = π‘šπ‘Žπ‘₯(0, 𝑋 βˆ’ 75)
variable.

(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(π‘₯)
, where𝐹(π‘₯) = 1
2 + 𝐺(π‘₯)
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 = π‘‹βˆ’ΞΌ
Οƒ
standard normal random variable, and are mean and varianceΞΌ Οƒ
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 .𝐹π‘₯(𝑠) = 𝑃(𝑋 < π‘₯ + 𝑠|𝑋 > 75)
(d) Consider (2) lives aged 75 with future lifetimes which following
Mortality assumptions:
𝑠𝑝π‘₯ = 𝑃(𝑋 > π‘₯ + 𝑠|𝑋 > 75)
𝑠𝑝𝑦 = 𝑃(π‘Œ > 𝑦 + 𝑠|π‘Œ > 75)
where and assume both and𝑋 ∼ 𝑁(75, 1) π‘Œ ∼ 𝑁(75, 1) 𝑋 π‘Œ
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 > 0𝑑𝑝’(1)
π‘₯ = π‘’βˆ’0.05𝑑
= Uniform distribution on the interval [0,45]𝑑𝑝’(2)
π‘₯
= , t > 0𝑑𝑝’(3)
π‘₯ π‘’βˆ’0.10𝑑
Where (1), (2), (3) represent Dismissal, withdrawal and retirement
respectively. Determine the following:
1. .𝑑𝑝 (Ο„)
π‘₯
2. π‘‘π‘ž(Ο„)
π‘₯ .
3. If A and B are independent show that and are𝐴 𝐡
independent (i.e. their complements are independent).
4. Hence, Prove the following relationship:
= +( +π‘‘π‘ž(𝑖)
π‘₯ π‘‘π‘ž‘(𝑖)
π‘₯𝑑𝑝‘(𝑗)
π‘₯𝑑𝑝‘(π‘˜)
π‘₯ π‘‘π‘ž‘(𝑖)
π‘₯π‘‘π‘ž‘(𝑗)
π‘₯𝑑𝑝‘(π‘˜)
π‘₯ π‘‘π‘ž‘(𝑖)
π‘₯𝑑𝑝‘(𝑗)
π‘₯
) +π‘‘π‘ž‘(π‘˜)
π‘₯ π‘‘π‘ž‘(𝑖)
π‘₯π‘‘π‘ž‘(𝑗)
π‘₯π‘‘π‘ž‘(π‘˜)
π‘₯

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)
π‘₯
Total: 15 marks