Question

It is said that a random variableX has an increasing failure rate if the failure rate h(x) defined in Exercise 18 is an increasing function of xfor x> 0, and it is said that Xhas a decreasing failure rate if h(x) is a decreasing function of x for x > 0. Suppose that X has the Weibull distribution with parameters a and b, as defined in Exercise 19. Show thatX has an increasing failure rate if b > 1, and X has a decreasing failure rate if b < 1.

Short answer

LetX has an increasing failure rate if b > 1, and X has a decreasing failure rate if b < 1.

Step-by-step solution

Given information

LetX has the Weibull distribution with parameters a and b.One need to prove that X has an increasing failure rate if b > 1, and X has a decreasing failure rate if b < 1.

Proof of X has an increasing failure rate if b > 1, and X has a decreasing failure rate if b < 1.

The failure rate of X is

\(h\left( x \right) = \frac{{f\left( x \right)}}{{1 - F\left( x \right)}}\)

Differentiating both sides with respect to x one get

\(\begin{array}{l}h'\left( x \right) = \frac{{f'\left( x \right)\left( {1 - F\left( x \right)} \right) - \left( { - f\left( x \right)} \right)f\left( x \right)}}{{{{\left\{ {1 - F\left( x \right)} \right\}}^2}}}\,\,{\rm{since}}\,F'\left( x \right) = f\left( x \right)\\\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{f'\left( x \right)\left( {1 - F\left( x \right)} \right) + {{\left( {f\left( x \right)} \right)}^2}}}{{{{\left\{ {1 - F\left( x \right)} \right\}}^2}}}\end{array}\)

Now,\(h'\left( x \right) > 0\,or\, < 0\,iff\,f'\left( x \right) > 0\,or\, < 0\).Now,

\(\begin{array}{l}f\left( x \right) = \frac{b}{{{a^b}}}{x^{b - 1}}{e^{ - {{\left( {\frac{x}{a}} \right)}^b}}}\\f'\left( x \right) = \frac{b}{{{a^b}}}\left( {\left( {b - 1} \right){x^{b - 2}}{e^{ - {{\left( {\frac{x}{a}} \right)}^b}}} + \frac{b}{{{a^b}}}{x^{b - 1}}{e^{ - {{\left( {\frac{x}{a}} \right)}^b}}}} \right)\\\,\,\,\,\,\,\,\,\,\,\,\,\, = \left( {b - 1} \right) \times k\,{\rm{where}}\,k > 0\\f'\left( x \right) > 0\,{\rm{if}}\,b > 1\,{\rm{and}}\, < 0\,if\,b < 1\end{array}\)

Hence X has an increasing failure rate if b > 1, and X has a decreasing failure rate if b < 1.