If T is a continuous random variable that is always positive (such as a waiting time), with probability density function f(t) and cumulative distribution function F(t), then the hazard function is defined to be the function

The hazard function is the rate of failure per unit time, expressed as a proportion of the items that have not failed.

a. If T~ Weibull (α, β), find h (i).

b. For what values of α is the hazard rate increasing with time? For what values of a is it decreasing?

c. If T has an exponential distribution, show that the hazard function is constant.

Step 1:

Let T is a continuous random variable that is always positive, with probability density function f(t) and cumulative distribution function F(t).The hazard function is the rate of failure per unit time which is defined as

h(t) =

Step 2:

We have to find the hazard function h(t), if T~weibull()If T~ weibull ( then its pdf will be in the form

f(T) = ,

And the cdf of T is

F(T) = 1-

1- F(T) =

The hazard function , h(t) =

=

=

Therefore the hazard function of T is ,h(t) = .