Math Homework
Math 5067 001 Homework 1 Due 9/11/13
1. Read Chapter 1 in the DHW text (sections 1.1 – 1.3 are mandatory) and answer the following: a. List at least three incentives for an insurance company to develop new insurance products. b. (Exercise 1.1 in DHW) Why do insurers generally require evidence of health from a person applying for life insurance but not for an annuity? c. (Exercise 1.3 in DHW) Explain why premiums are payable in advance, so that the first premium is due at issue, rather than in one year’s time.
2. Let ������! ������ = 1 − (1 − !"#)! for 0 ≤ ������ ≤ 105. Calculate: a. The probability that a newborn life dies before age 60. b. The probability that a life aged 30 survives to at least age 70. c. The probability that a life aged 20 dies between ages 90 and 100
18000 −110x − x 2 has been proposed as the survival function ������! (������) for a 18000
!
!
3. The function G(x) =
mortality model. a. Under which conditions G(x) satisfy the criteria for a survival function? b. Determine the survival function for life aged 20. c. Calculate the probability that a life aged 20 dies between ages 30 and 40 d. Calculate 20 p0 . 4. Show that if X is a random variable such that P(X ≥ 0) = 1 then
∞
a. E[X] = €
∫ s(x)dx
0 ∞ 0
€
b. E[X 2 ] = 2 ∫ xs(x)dx where s(x) is the survival function for X .
€ 5. Find the expected value E[X] and the variance Var(X) for the following random variables ( X ):
a.
X for which µ (x) = 0.5 for x ≥ 0 . €
€ €
x € b. X for which the CDF F(x) = € for 0 ≤ x ≤ 100 . 100 €
€
6. Given that px = 0.99 , px +1 = 0.985 , 3 px +1 = 0.95 and qx +3 = 0.02 , calculate: a. px +3 b. 2 px c. 2 px +1 € € € € d. 3 px € e. 1|2 qx € € € €7. Show that q = p q t|u x t x u x+t . 8. Given that the force of mortality µx = 2x , determine the cumulative distribution function for the random variable age at death, FX (x) , the probability density function f X (x) , and the survival function sX (x) .
€ € €
d 9. €
1. Read Chapter 1 in the DHW text (sections 1.1 – 1.3 are mandatory) and answer the following: a. List at least three incentives for an insurance company to develop new insurance products. b. (Exercise 1.1 in DHW) Why do insurers generally require evidence of health from a person applying for life insurance but not for an annuity? c. (Exercise 1.3 in DHW) Explain why premiums are payable in advance, so that the first premium is due at issue, rather than in one year’s time.
2. Let ������! ������ = 1 − (1 − !"#)! for 0 ≤ ������ ≤ 105. Calculate: a. The probability that a newborn life dies before age 60. b. The probability that a life aged 30 survives to at least age 70. c. The probability that a life aged 20 dies between ages 90 and 100
18000 −110x − x 2 has been proposed as the survival function ������! (������) for a 18000
!
!
3. The function G(x) =
mortality model. a. Under which conditions G(x) satisfy the criteria for a survival function? b. Determine the survival function for life aged 20. c. Calculate the probability that a life aged 20 dies between ages 30 and 40 d. Calculate 20 p0 . 4. Show that if X is a random variable such that P(X ≥ 0) = 1 then
∞
a. E[X] = €
∫ s(x)dx
0 ∞ 0
€
b. E[X 2 ] = 2 ∫ xs(x)dx where s(x) is the survival function for X .
€ 5. Find the expected value E[X] and the variance Var(X) for the following random variables ( X ):
a.
X for which µ (x) = 0.5 for x ≥ 0 . €
€ €
x € b. X for which the CDF F(x) = € for 0 ≤ x ≤ 100 . 100 €
€
6. Given that px = 0.99 , px +1 = 0.985 , 3 px +1 = 0.95 and qx +3 = 0.02 , calculate: a. px +3 b. 2 px c. 2 px +1 € € € € d. 3 px € e. 1|2 qx € € € €7. Show that q = p q t|u x t x u x+t . 8. Given that the force of mortality µx = 2x , determine the cumulative distribution function for the random variable age at death, FX (x) , the probability density function f X (x) , and the survival function sX (x) .
€ € €
d 9. €