Template:Lognormal distribution probability plotting

Probability Plotting
As described before, probability plotting involves plotting the failure times and associated unreliability estimates on specially constructed probability plotting paper. The form of this paper is based on a linearization of the $$cdf$$  of the specific distribution. For the lognormal distribution, the cumulative density function can be written as:


 * $$F({t}')=\Phi \left( \frac{{t}'-{\mu }'} \right)$$

or:


 * $${{\Phi }^{-1}}\left[ F({t}') \right]=-\frac+\frac{1}\cdot {t}'$$

where:


 * $$\Phi (x)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{x}{{e}^{-\tfrac{2}}}dt$$

Now, let:


 * $$y={{\Phi }^{-1}}\left[ F({t}') \right]$$


 * $$a=-\frac$$

and:


 * $$b=\frac{1}$$

which results in the linear equation of:


 * $$y=a+b{t}'$$

The normal probability paper resulting from this linearized $$cdf$$  function is shown next.

The process for reading the parameter estimate values from the lognormal probability plot is very similar to the method employed for the normal distribution (see The Normal Distribution Chapter). However, since the lognormal distribution models the natural logarithms of the times-to-failure, the values of the parameter estimates must be read and calculated based on a logarithmic scale, as opposed to the linear time scale as it was done with the normal distribution. This parameter scale appears at the top of the lognormal probability plot.

The process of lognormal probability plotting is illustrated in the following example.

Example 1: