Question

Let the random variable T be the time until some event occurs (e.g. time until an atom decays, time until next rainfall, etc). Suppose it’s a continuous random variable supported on [0, ∞). The hazard rate for T is defined as h(t) = f(t) 1 − F(t) , where f and F are the density and CDF for the distribution of T. On an intuitive level, this is the chance that the event will occur in the very near future, given that it has not yet occurred. Hazard rate is a function of time since it can rise or fall as time goes on. (a) Calculate the hazard rate h(t) if T ∼ Exponential(λ) (b) Now suppose that T follows a Weibull distribution with shape k > 0 and scale α > 0, which is defined by the CDF F(t) = 1 − e −(t/α) k . Calculate the density f(t), and the hazard rate function h(t), for this distribution. (c) For the Weibull distribution, for which values of k and α is h(t) decreasing over time, increasing over time, or constant over time? (d) Next, suppose that you buy a new watch. When you purchase it, you put in a new battery. Let T be defined as the time from putting in the new battery, until the battery runs out. Do you think the hazard rate function for T should be decreasing over time, increasing over time, or constant over time (or some other shape)? Explain. (There may be multiple plausible answers.) (e) Finally, a common shape for the hazard function is a “U” shape, where the hazard rate is high initially, goes down to a low rate for a long time, and then rises again later on. Give a real-life example for what T could measure that would likely have this type of hazard rate function, and explain.

Answer 1

(a)

T ∼ Exponential(λ)

For exponential distribution,

Hazard rate =

(b)

T follows a Weibull distribution with shape k > 0 and scale α > 0

For Weibull distribution,

Hazard rate =

(c)

For k > 1 and t > , hazard rate increases with time. For k > 1 and t < , hazard rate decreases with time.

For k = 1 hazard rate is constant over time.

For 0 < k < 1 and and t > , hazard rate decreases with time. For k > 1 and t < , hazard rate increases with time.

(d)

The common shape for the hazard function is a “U” shape. The hazard rate function decreases over time initially and then is constant for long time and then increases over time until the battery runs out.