Section 3.1 Class Handout

Section 3.1 Class Handout

Math 400 – Actuarial Models NAME:______

1. Let l(x) = abxb - 1 for 0 £ x, where a and b are constants.

(a) Find the values of a and b for which l(x) could be the hazard rate for a survival distribution.

(b) Suppose the values of a and b are such that l(x) is the hazard rate for a survival distribution. Find the survival function SX(x), the p.d.f. fX(x), and the mode of the distribution.

(c) Suppose b = 1. Name the type of distribution that the lifetime random variable X has, and state what the mean of X is.

2. Let l(x) = for 0 £ x £ 1/a , where a is a constant.

(a) Find the values of a for which l(x) could be the hazard rate for a survival distribution.

(b) Suppose the value of a is such that l(x) is the hazard rate for a survival distribution. Find the survival function SX(x), find the p.d.f. fX(x), name the type of distribution that the lifetime random variable X has, and state what the mean of X is.

3. Let l(x) = ex for 0 £ x. Decide whether or not l(x) could be the hazard rate for a survival distribution. If not, demonstrate why; if yes, find the survival function SX(x), and find the p.d.f. fX(x).

4. Demonstrate that l(x) = e - x for 0 £ x cannot be the hazard rate for a survival distribution.

5. Demonstrate that each of the functions listed, except for one, cannot be a survival function.

(a) cos(x) for 0 £ x £ 3p/2

(b) cos(x) for 0 £ x £ p/2

(d) for 0 £ x £ 3p

6. Let SX(x) = axn + b for 0 £ x £ k , where a, b, and k are constants, and n is a positive constant.

(a) Find the value of b.

(b) Find a in terms of k and n.

(d) Find E(X) in terms of k and n.

(e) If n = 1, name the type of distribution that the lifetime random variable X has.

7. Let SX(x) = (ax + b) n for 0 £ x £ k , where a, b, and k are constants, and n is a positive constant.

(a) Find the value of b.

(b) Find a in terms of k and n.

(d) Find E(X) in terms of k and n.

(e) If n = 1, name the type of distribution that the lifetime random variable X has.