Template:Alta statistical properties summary

The Mode
The mode, $$\tilde{T},$$ of the 1-parameter exponential distribution is given by:
 * $$\tilde{T}=0$$

The Standard Deviation
The standard deviation, $${{\sigma }_{T}}$$, of the 1-parameter exponential distribution is given by:
 * $${{\sigma }_{T}}=\frac{1}{\lambda }=m$$

The Reliability Function
The 1-parameter exponential reliability function is given by:
 * $$R(T)={{e}^{-\lambda T}}={{e}^{-\tfrac{T}{m}}}$$

This function is the complement of the exponential cumulative distribution function or:
 * $$R(T)=1-Q(T)=1-\mathop{}_{0}^{T}f(T)dT$$


 * and:


 * $$R(T)=1-\mathop{}_{0}^{T}\lambda {{e}^{-\lambda T}}dT={{e}^{-\lambda T}}$$

Conditional Reliability
The conditional reliability function for the 1-parameter exponential distribution is given by:
 * $$R(T,t)=\frac{R(T+t)}{R(T)}=\frac={{e}^{-\lambda t}}$$

which says that the reliability for a mission of $$t$$  duration undertaken after the component or equipment has already accumulated  $$T$$  hours of operation from age zero is only a function of the mission duration, and not a function of the age at the beginning of the mission. This is referred to as the ``memoryless property.''

Reliable Life
The reliable life, or the mission duration for a desired reliability goal, $${{t}_{R}}$$, for the 1-parameter exponential distribution is given by:


 * $$\begin{align}

& R({{t}_{R}})= & {{e}^{-\lambda {{t}_{R}}}} \\ & &  \\  & \ln [R({{t}_{R}})]= & -\lambda {{t}_{R}} \end{align}$$
 * or:


 * $${{t}_{R}}=-\frac{\ln [R({{t}_{R}})]}{\lambda }$$

Failure Rate Function
The exponential failure rate function is given by:
 * $$\lambda (T)=\frac{f(T)}{R(T)}=\frac{\lambda {{e}^{-\lambda (T)}}}=\lambda =\text{Constant}$$