This is part of a short series on the common life data distributions.
The Lognormal distribution is a versatile and continuous distribution. It is similar to the Weibull in flexibility with just slightly fatter tails in most circumstances. It is commonly used to describe time to repair behavior. This short article focuses on 7 formulas of the Lognormal Distribution.
If you want to know more about fitting a set of data to a distribution, well that is in another article.
It has the essential formulas that you may find useful when answering specific questions. Knowing a distribution’s set of parameters does provide, along with the right formulas, a quick means to answer a wide range of reliability related questions.
Parameters
The scale parameter, μN is the mean of the normally distributed natural logarithm of the data, ln(x). Unlike the normal distribution this parameter is only the scale and not the location. The scale parameter ranges from -∞ < μN < ∞ and if found from the data with: $$ \displaystyle\large {{\mu }_{N}}=\ln \left( \frac{{{\mu }^{2}}}{\sqrt{{{\sigma }^{2}}+{{\mu }^{2}}}} \right)$$ The shape parameter, σ2N is the standard deviation of the normally distributed ln(x). Unlike the normal distribution this parameter is only the shape and not the scale. The shape parameter is always positive and is determined by the data using: $$ \displaystyle\large \sigma _{N}^{2}=\ln \left( \frac{{{\sigma }^{2}}+{{\mu }^{2}}}{{{\mu }^{2}}} \right)$$ In summary despite the parameters being known as sigma and mu they are not the mean and standard deviation of the distribution, thus be cautious interpreting the parameters. Probability Density Function (PDF) When t > 0 then the probability density function formula is:
$$ \displaystyle\large \begin{array}{l}f(t)=\frac{1}{{{\sigma }_{N}}t\sqrt{2\pi }}\exp \left[ -\frac{1}{2}{{\left( \frac{\ln \left( t \right)-{{\mu }_{N}}}{{{\sigma }_{N}}} \right)}^{2}} \right]\\f(t)=\frac{1}{{{\sigma }_{N}}t}\phi \left[ \frac{\ln \left( t \right)-{{\mu }_{N}}}{{{\sigma }_{N}}} \right]\end{array}$$
Where ? is the standard normal PDF.
A plot of the PDF provides a histogram-like view of the time-to-failure data.
Cumulative Density Function (CDF)
F(t) is the cumulative probability of failure from time zero till time t. Very handy when estimating the proportion of units that will fail over a warranty period, for example.
$$ \displaystyle\large \begin{array}{l}F(t)=\frac{1}{{{\sigma }_{N}}t\sqrt{2\pi }}\int_{0}^{t}{\exp \left[ -\frac{1}{2}{{\left( \frac{\ln \left( t \right)-{{\mu }_{N}}}{{{\sigma }_{N}}} \right)}^{2}} \right]}dt\\F(t)=\Phi \left[ \frac{\ln \left( t \right)-{{\mu }_{N}}}{{{\sigma }_{N}}} \right]\end{array}$$
Where Φ is the standard normal CDF.
Reliability Function
R(t) is the chance of survival from time zero till time t. Instead of looking for the proportion that will fail the reliability function determine the proportion that is expected to survive.
$$ \displaystyle\large R(t)=1-\Phi \left[ \frac{\ln \left( t \right)-{{\mu }_{N}}}{{{\sigma }_{N}}} \right]$$
Conditional Survivor Function
The m(x) function provides a means to estimate the chance of survival for a duration beyond some known time, t, over which the item(s) have already survived. What is the probability of surviving time x given the item has already survived over time t?
$$ \displaystyle\large m(x)=R\left( x|t \right)=\frac{1-\Phi \left[ \frac{\ln \left( x+t \right)-{{\mu }_{N}}}{{{\sigma }_{N}}} \right]}{1-\Phi \left[ \frac{\ln \left( t \right)-{{\mu }_{N}}}{{{\sigma }_{N}}} \right]}$$
Mean Residual Life
This is the cumulative expected life over time x given survival till time t.
$$ \displaystyle\large u(t)=\frac{\int_{t}^{\infty }{R\left( x \right)dx}}{R\left( t \right)}$$
Hazard Rate
This is the instantaneous probability of failure per unit time.
$$ \displaystyle\large h(x)=\frac{\phi \left[ \frac{\ln \left( t \right)-{{\mu }_{N}}}{{{\sigma }_{N}}} \right]}{t{{\sigma }_{N}}\left( 1-\left[ \frac{\ln \left( t \right)-{{\mu }_{N}}}{{{\sigma }_{N}}} \right] \right)}$$
Cumulative Hazard Rate
This is the cumulative failure rate from time zero till time t, or the area under the curve described by the hazard rate, h(x).
$$ \displaystyle\large H\left( t \right)=-\ln \left[ R\left( t \right) \right]$$
Recent probability distribution case study is needed.