Template:Exponential Statistical Properties

The Mean or MTTF
The mean, $$\overline{T},$$ or mean time to failure (MTTF) is given by:


 * $$\begin{align}

\bar{T}= & \int_{\gamma }^{\infty }t\cdot f(t)dt \\ = & \int_{\gamma }^{\infty }t\cdot \lambda \cdot {{e}^{-\lambda t}}dt \\ = & \gamma +\frac{1}{\lambda }=m \end{align}$$

Note that when $$\gamma =0$$, the MTTF is the inverse of the exponential distribution's constant failure rate. This is only true for the exponential distribution. Most other distributions do not have a constant failure rate. Consequently, the inverse relationship between failure rate and MTTF does not hold for these other distributions.