The Normal Distribution

The normal distribution, also known as the Gaussian distribution, is the most widely-used general purpose distribution. It is for this reason that it is included among the lifetime distributions commonly used for reliability and life data analysis. There are some who argue that the normal distribution is inappropriate for modeling lifetime data because the left-hand limit of the distribution extends to negative infinity. This could conceivably result in modeling negative times-to-failure. However, provided that the distribution in question has a relatively high mean and a relatively small standard deviation, the issue of negative failure times should not present itself as a problem. Nevertheless, the normal distribution has been shown to be useful for modeling the lifetimes of consumable items, such as printer toner cartridges.

Normal Probability Density Function
The pdf of the normal distribution is given by:


 * $$f(t)=\frac{1}{\sigma \sqrt{2\pi }}{{e}^{-\frac{1}{2}{{\left( \frac{t-\mu }{\sigma } \right)}^{2}}}}\,\!$$

where:


 * $$\mu\,\!$$ = mean of the normal times-to-faiure, also noted as $$\bar{T}\,\!$$,


 * $$\theta\,\!$$ = standard deviation of the times-to-failure

It is a 2-parameter distribution with parameters $$\mu \,\!$$ (or $$\bar{T}\,\!$$ ) and $$\,\!$$ (i.e., the mean and the standard deviation, respectively).

The Normal Mean, Median and Mode
The normal mean or MTTF is actually one of the parameters of the distribution, usually denoted as $$\mu .\,\!$$ Because the normal distribution is symmetrical, the median and the mode are always equal to the mean:


 * $$\mu =\tilde{T}=\breve{T}\,\!$$

The Normal Standard Deviation
As with the mean, the standard deviation for the normal distribution is actually one of the parameters, usually denoted as $${{\sigma }_{T}}\,\!$$.

The Normal Reliability Function
The reliability for a mission of time $$T\,\!$$ for the normal distribution is determined by:


 * $$R(t)=\int_{t}^{\infty }f(x)dx=\int_{t}^{\infty }\frac{1}{{{\sigma }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{x-\mu }{{{\sigma }}} \right)}^{2}}}}dx\,\!$$

There is no closed-form solution for the normal reliability function. Solutions can be obtained via the use of standard normal tables. Since the application automatically solves for the reliability, we will not discuss manual solution methods. For interested readers, full explanations can be found in the references.

The Normal Conditional Reliability Function
The normal conditional reliability function is given by:


 * $$R(t|T)=\frac{R(T+t)}{R(T)}=\frac{\int_{T+t}^{\infty }\tfrac{1}{{{\sigma }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{x-\mu }{{{\sigma }}} \right)}^{2}}}}dx}{\int_{T}^{\infty }\tfrac{1}{{{\sigma }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{x-\mu }{{{\sigma }}} \right)}^{2}}}}dx}\,\!$$

Once again, the use of standard normal tables for the calculation of the normal conditional reliability is necessary, as there is no closed form solution.

The Normal Reliable Life
Since there is no closed-form solution for the normal reliability function, there will also be no closed-form solution for the normal reliable life. To determine the normal reliable life, one must solve:


 * $$R(T)=\int_{T}^{\infty }\frac{1}{{{\sigma }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\mu }{{{\sigma }}} \right)}^{2}}}}dt\,\!$$

for $$T\,\!$$.

The Normal Failure Rate Function
The instantaneous normal failure rate is given by:


 * $$\lambda (t)=\frac{f(t)}{R(t)}=\frac{\tfrac{1}{{{\sigma }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\mu }{{{\sigma }}} \right)}^{2}}}}}{\int_{t}^{\infty }\tfrac{1}{{{\sigma }}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{x-\mu }{{{\sigma }}} \right)}^{2}}}}dx}\,\!$$

Characteristics of the Normal Distribution
Some of the specific characteristics of the normal distribution are the following:
 * The normal pdf has a mean, $$\bar{T}\,\!$$, which is equal to the median, $$\breve{T}\,\!$$, and also equal to the mode, $$\tilde{T}\,\!$$, or $$\bar{T}=\breve{T}=\tilde{T}\,\!$$. This is because the normal distribution is symmetrical about its mean.




 * The mean, $$\mu \,\!$$, or the mean life or the $$MTTF\,\!$$, is also the location parameter of the normal pdf, as it locates the pdf along the abscissa. It can assume values of $$-\infty <\bar{T}<\infty \,\!$$.
 * The normal pdf has no shape parameter. This means that the normal pdf has only one shape, the bell shape, and this shape does not change.




 * The standard deviation, $$\,\!$$, is the scale parameter of the normal pdf.


 * As $$\,\!$$ decreases, the pdf gets pushed toward the mean, or it becomes narrower and taller.


 * As $$\,\!$$ increases, the pdf spreads out away from the mean, or it becomes broader and shallower.


 * The standard deviation can assume values of $$0<<\infty \,\!$$.


 * The greater the variability, the larger the value of $$\,\!$$, and vice versa.


 * The standard deviation is also the distance between the mean and the point of inflection of the pdf, on each side of the mean. The point of inflection is that point of the pdf where the slope changes its value from a decreasing to an increasing one, or where the second derivative of the pdf has a value of zero.


 * The normal pdf starts at $$t=-\infty \,\!$$ with an $$f(t)=0\,\!$$. As $$t\,\!$$ increases, $$f(t)\,\!$$ also increases, goes through its point of inflection and reaches its maximum value at $$t=\bar{T}\,\!$$. Thereafter, $$f(t)\,\!$$ decreases, goes through its point of inflection, and assumes a value of $$f(t)=0\,\!$$ at $$t=+\infty \,\!$$.

Weibull++ Notes on Negative Time Values

One of the disadvantages of using the normal distribution for reliability calculations is the fact that the normal distribution starts at negative infinity. This can result in negative values for some of the results. Negative values for time are not accepted in most of the components of Weibull++, nor are they implemented. Certain components of the application reserve negative values for suspensions, or will not return negative results. For example, the Quick Calculation Pad will return a null value (zero) if the result is negative. Only the Free-Form (Probit) data sheet can accept negative values for the random variable (x-axis values).