Math 3280 Full package

Final test MATH 3280 3.00 December 6, 2019


Given name and surname:

Student No:

Signature:




INSTRUCTIONS:

1. Please write everything in ink.

2. This exam is a ‘closed book’ test, duration 180 minutes.

3. Only non-programmable calculators are permitted.

4. There are fourteen questions and a bonus question.




USEFUL FORMULAS:

For x ≥ 0, t ∈ [0, 1) and k = 0, 1, 2, . . ., if the uniform distribution of
deaths assumption holds for the life-status (x), then the following is true

t+k px ≈ (1 − t)k px + tk+1 px .


Let C : [0, 1]2 → [0, 1] denote a copula function, and let u, v ∈ [0, 1],
then the Fréchet-Hoeffding bounds state

max (u + v − 1, 0) ≤ C(u, v) ≤ min(u, v).




GOOD LUCK!

,Final test MATH 3280 3.00 Page 2 of 30

1
1. Recall that we denote by (u) a general life-status. Choose (u) = (x : n ), x ≥ 0, n =
1, 2, . . .

• What insurance contract does this life-status correspond to? Explain in one
sentence.
1
• Write
 1formally the random variable T (u) = T (x : n ) as well as the probability
P T (x : n ) ≥ t , t ≥ 0.
• Assume that the life-status (x) admits the uniform distribution of deaths approx-
1
imation, check whether the life-status (x : n ) also admits the uniform distribution
of deaths approximation.




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, Final test MATH 3280 3.00 Page 3 of 30


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