Instructions:
β Assignment consists of 2 questions. Attempt all questions.
β Questions are to be compiled into one document.
β Questions can be written or typed.
β Written questions can be scanned and compiled into a single pdf
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β 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:
.π ππ₯ = π(π > π₯ + π |π > 75)
i.e. X follows a normal distribution with meanπ βΌ π(75, 1)
75 and variance 1.
, where S is the future lifetime randomπ = πππ₯(0, π β 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:πΉ(π₯)
, whereπΉ(π₯) = 1
2 + πΊ(π₯)
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(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 andπ βΌ π(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 aπ = πβΞΌ
Ο
standard normal random variable, and are mean and varianceΞΌ Ο
of X respectively where .π βΌ π(75, 1)
Adjusted Approx Normal Cumulative Probabilities: Hint: to
Survival function: use Adjusted Approx Normal Cumulative
Probabilities to calculate.
make the adjustment compute .πΉπ₯(π ) = π(π < π₯ + π |π > 75)
(d) Consider (2) lives aged 75 with future lifetimes which following
Mortality assumptions:
π ππ₯ = π(π > π₯ + π |π > 75)
π ππ¦ = π(π > π¦ + π |π > 75)
where and assume both andπ βΌ π(75, 1) π βΌ π(75, 1) π π
are independent.Using the results from (c) Calculate:
and (survival probabilities for 1 year for both theπ75:75 π75:75
joint and last survivor status respectively).
Total: 20 marks
Question 2:
Consider the following ASD (Associated Single Decrement) survival
functions for (x):
, t > 0π‘π’(1)
π₯ = πβ0.05π‘
= Uniform distribution on the interval [0,45]π‘π’(2)
π₯
= , t > 0π‘π’(3)
π₯ πβ0.10π‘
Where (1), (2), (3) represent Dismissal, withdrawal and retirement
respectively. Determine the following:
1. .π‘π (Ο)
π₯
2. π‘π(Ο)
π₯ .
3. If A and B are independent show that and areπ΄‘ π΅‘
independent (i.e. their complements are independent).
4. Hence, Prove the following relationship:
= +( +π‘π(π)
π₯ π‘π‘(π)
π₯π‘π‘(π)
π₯π‘π‘(π)
π₯ π‘π‘(π)
π₯π‘π‘(π)
π₯π‘π‘(π)
π₯ π‘π‘(π)
π₯π‘π‘(π)
π₯
) +π‘π‘(π)
π₯ π‘π‘(π)
π₯π‘π‘(π)
π₯π‘π‘(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 π(3)
π₯
Total: 15 marks