Log-normal Distribution

The log-normal distribution can be a good fit to the distribution of component repair times and consequently is becoming an important distribution in the analysis of the repairable systems. The PDF for the distribution is

Where σ = log standard deviation of t.

X = log10 t;

ξ = log mean value of x;

0.4343 =1 / log₁₀

Log-normal cumulative distribution function is following

F (t) = φ( log - (ξ / σ ))                                 

where "φ " is the standard normal cumulative distribution function and can be given by

The log-normal reliability function is specified by

R (t) = 1 - φ( log  (- ξ / σ );   t > 0