# Is it true that for the Weibull distribution a value of $k$ for $k > 1$ indicates that the failure rate increases with time?

When describing the Weibull distribution on Wikipedia, it is claimed, among other things, that

"A value of $$k > 1$$ indicates that the failure rate increases with time."

However, for $$k > 3$$ the failure rate increases with time, but then decreases after a while.

For example for $$k = 4$$ the failure rate $$h(x;\lambda,4)$$ is as follows:

$$h \left(x \right) = \frac{4~x ^{3}~e^{\frac{x ^{3}~\left(\lambda -x \right)}{\lambda ^{4}}}}{\lambda ^{4}}$$

Is this correct?

The failure rate is $$h(x)=f(x)/(1-F(x))$$, and when substituted it is $$h(x)=\frac{\frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}e^{-(x/\lambda)^k}}{1-(1-e^{-(x/\lambda)^k})}=\frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}$$ which increases as $$x$$ increases if $$k>1$$.
• I am very, very sorry for making an error in the derivation of $h(x)$. – Ad van der Ven Nov 2 '19 at 12:11