Suppose the random variable X is a duration (a period of time). Examples include: a spell of unemployment, length of time on a job, length of a labor strike, length of a recession, length of an economic expansion. The hazard function h(x) associated with X is h(x) Æ lim

±!0 P[x · X · x ű j X ¸ x]

±

.

This can be interpreted as the rate of change of the probability of continued survival. If h(x) increases

(decreases) with x this is described as increasing (decreasing) hazard.

(a) Show that if X has distribution F and density f then h(x) Æ f (x)/ (1¡F(x)) .

(b) Suppose f (x) Æ ¸¡1 exp(¡x/¸). Find the hazard function h(x). Is this increasing or decreasing hazard or neither?

(c) Find the hazard function for the Weibull distribution from Section 3.23. Is this increasing or decreasing hazard or neither?