Template: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.

General Examples
Example 8:

Example 9:

Example 10:

Example 11
Eight units are being reliability tested and the following is a table of their times-to-failure:

Solution to Example 11
This is a sequence of interval times-to-failure. This data set can be entered into Weibull++ by creating a data sheet that can be used to analyze times-to-failure data with interval and left censored data.

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The computed parameters for maximum likelihood are:


 * $$\begin{align}

& \widehat{\mu }= & 41.40 \\ & {{{\hat{\sigma }}}_{T}}= & 7.740. \end{align}$$

For rank regression on x:


 * $$\begin{align}

& \widehat{\mu }= & 41.40 \\ & {{{\hat{\sigma }}}_{T}}= & 9.03. \end{align}$$

For rank regression on y:


 * $$\begin{align}

& \widehat{\mu }= & 41.39 \\ & {{{\hat{\sigma }}}_{T}}= & 9.25. \end{align}$$

A plot of the MLE solution is shown next.

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Example 12
Eight units are being reliability tested and the following is a table of their times-to-failure:

Solution to Example 12
This data set can be entered into Weibull++ by creating a Data Sheet appropriate for the entry of non-grouped times-to-failure data. The computed parameters for maximum likelihood are:


 * $$\begin{align}

& \widehat{\mu }= & 26.13 \\ & {{{\hat{\sigma }}}_{T}}= & 18.57 \end{align}$$

For rank regression on x:


 * $$\begin{align}

& \widehat{\mu }= & 26.13 \\ & {{{\hat{\sigma }}}_{T}}= & 21.64 \end{align}$$

For rank regression on y:


 * $$\begin{align}

& \widehat{\mu }= & 26.13 \\ & {{{\hat{\sigma }}}_{T}}= & 22.28. \end{align}$$

Example 13
Nineteen units are being reliability tested and the following is a table of their times-to-failure and suspensions.

Solution to Example 13
This augments the previous example by adding eleven suspensions to the data set. This data set can be entered into Weibull++ by selecting the data sheet for Times to Failure and with Right Censored Data (Suspensions). The parameters using maximum likelihood are:


 * $$\begin{align}

& \widehat{\mu }= & 48.07 \\ & {{{\hat{\sigma }}}_{T}}= & 28.41. \end{align}$$

For rank regression on x:


 * $$\begin{align}

& \widehat{\mu }= & 46.40 \\ & {{{\hat{\sigma }}}_{T}}= & 28.64. \end{align}$$

For rank regression on y:


 * $$\begin{align}

& \widehat{\mu }= & 47.34 \\ & {{{\hat{\sigma }}}_{T}}= & 29.96. \end{align}$$

Example 14
Suppose our data set includes left and right censored, interval censored and complete data as shown in the following table.

Solution to Example 14
This data set can be entered into Weibull++ by selecting the data type Times to Failure, with Right Censored Data (Suspensions), with Interval and Left Censored Data and with Grouped Observations.

The computed parameters using maximum likelihood are:


 * $$\begin{align}

& \widehat{\mu }= & 48.11 \\ & {{{\hat{\sigma }}}_{T}}= & 26.42. \end{align}$$

For rank regression on x:


 * $$\begin{align}

& \widehat{\mu }= & 49.99 \\ & {{{\hat{\sigma }}}_{T}}= & 30.17. \end{align}$$

For rank regression on y:


 * $$\begin{align}

& \widehat{\mu }= & 51.61 \\ & {{{\hat{\sigma }}}_{T}}= & 33.07. \end{align}$$