Template:Alta ld statistical prop func

The Mode

 * •	The mode of the lognormal distribution is given by:


 * $$\tilde{T}={{e}^{{{\bar{T}}^{\prime }}-\sigma _^{2}}}$$

Reliability Function
For the lognormal distribution, the reliability for a mission of time $$T$$, starting at age 0, is given by:
 * $$R(T)=\mathop{}_{T}^{\infty }f(t)dt$$


 * or:


 * $$R(T)=\mathop{}_^{\infty }\frac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\overline{{{T}'}}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt$$

There is no closed form solution for the lognormal reliability function. Solutions can be obtained via the use of standard normal tables.

Lognormal Failure Rate
The lognormal failure rate is given by:
 * $$\lambda (T)=\frac{f(T)}{R(T)}=\frac{\tfrac{1}{{T}'{{\sigma }_}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{(\tfrac{{T}'-\overline}{{{\sigma }_}})}^{2}}}}}{\mathop{}_^{\infty }\tfrac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{(\tfrac{t-\overline{{{T}'}}}{{{\sigma }_{{{T}'}}}})}^{2}}}}dt}$$