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{{template:LDABOOK|9|The Normal (Gaussian) Distribution}}
#REDIRECT [[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}}
 
{{normal statistical properties}}
 
{{characteristics of the normal distribution}}
 
{{normal distribution estimation of the parameters}}
 
{{normal distribution confidence bounds}}
 
{{nd bayesian confidence bounds}}
 
==General Examples==
 
'''Example 8:'''
{{Example: Normal General Example (RRX Plot)}}
 
 
'''Example 9:'''
{{Example: Normal General Example (RRX QCP)}}
 
 
'''Example 10:'''
{{Example: Normal General Example (RRX Report)}}
 
===Example 11===
Eight units are being reliability tested and the following is a table of their times-to-failure:
 
 
{|align="center" border=1 cellspacing=1
|-
|colspan="3" style="text-align:center"| Table 8.3 - Non-Grouped Data Times-to-Failure with intervals (lnterval and left censored)
|-
!Data point index
!Last Inspected
!State End Time
|-
|1 ||30||32
|-
|2 ||32||35
|-
|3 ||35||37
|-
|4 ||37||40
|-
|5 ||42||42
|-
|6 ||45||45
|-
|7||50||50
|-
|8||55||55
|}
 
====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.
 
<math></math>
[[Image:lastinspected.png|thumb|center|400px| ]]  
 
[[Image:lastinspectedsheet.png|thumb|center|400px]]
 
The computed parameters for maximum likelihood are:
 
::<math>\begin{align}
  & \widehat{\mu }= & 41.40 \\
& {{{\hat{\sigma }}}_{T}}= & 7.740. 
\end{align}</math>
 
For rank regression on x:
 
::<math>\begin{align}
  & \widehat{\mu }= & 41.40 \\
& {{{\hat{\sigma }}}_{T}}= & 9.03. 
\end{align}</math>
 
For rank regression on y:
 
::<math>\begin{align}
  & \widehat{\mu }= & 41.39 \\
& {{{\hat{\sigma }}}_{T}}= & 9.25. 
\end{align}</math>
 
 
A plot of the MLE solution is shown next.
 
<math></math>
[[Image:lastinspectedplot.png|thumb|center|400px| ]]
 
===  Example 12===
Eight units are being reliability tested and the following is a table of their times-to-failure:
 
 
{|align="center" border=1 cellspacing=1
|-
|colspan="3" style="text-align:center"| Table 8.4 - Non-Grouped Data ,for Example 12
|-
!Data point index
!State F or S
!State End Time
|-
|1 ||F||2
|-
|2 ||F||5
|-
|3 ||F||11
|-
|4 ||F||23
|-
|5 ||F||29
|-
|6 ||F||37
|-
|7||F||43
|-
|8||F||59
|}
 
==== 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:
 
::<math>\begin{align}
  & \widehat{\mu }= & 26.13 \\
& {{{\hat{\sigma }}}_{T}}= & 18.57 
\end{align}</math>
 
 
For rank regression on x:
 
::<math>\begin{align}
  & \widehat{\mu }= & 26.13 \\
& {{{\hat{\sigma }}}_{T}}= & 21.64 
\end{align}</math>
 
 
For rank regression on y:
 
::<math>\begin{align}
  & \widehat{\mu }= & 26.13 \\
& {{{\hat{\sigma }}}_{T}}= & 22.28. 
\end{align}</math>
 
===  Example 13===
Nineteen units are being reliability tested and the following is a table of their times-to-failure and suspensions.
 
{|align="center" border=1 cellspacing=1
|-
|colspan="3" style="text-align:center"| Table 8.5 - Non-Grouped Data Times-to-Failure with Suspensions (Right Censored)
|-
!Data point index
!Last Inspected
!State End Time
|-
|1 ||F||2
|-
|2 ||S||3
|-
|3 ||F||5
|-
|4 ||S||7
|-
|5 ||F||11
|-
|6 ||S||13
|-
|7||S||17
|-
|8||S||19
|-
|9 ||F||23
|-
|10 ||F||29
|-
|11 ||S||31
|-
|12 ||F||37
|-
|13 ||S||41
|-
|14 ||F||43
|-
|15||S||47
|-
|16||S||53
|-
|17||F||59
|-
|18||S||61
|-
|19||S||67
|}
 
====  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:
 
::<math>\begin{align}
  & \widehat{\mu }= & 48.07 \\
& {{{\hat{\sigma }}}_{T}}= & 28.41. 
\end{align}</math>
 
 
For rank regression on x:
 
::<math>\begin{align}
  & \widehat{\mu }= & 46.40 \\
& {{{\hat{\sigma }}}_{T}}= & 28.64. 
\end{align}</math>
 
 
For rank regression on y:
 
::<math>\begin{align}
  & \widehat{\mu }= & 47.34 \\
& {{{\hat{\sigma }}}_{T}}= & 29.96. 
\end{align}</math>
 
===  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:
 
::<math>\begin{align}
  & \widehat{\mu }= & 48.11 \\
& {{{\hat{\sigma }}}_{T}}= & 26.42. 
\end{align}</math>
 
 
For rank regression on x:
 
::<math>\begin{align}
  & \widehat{\mu }= & 49.99 \\
& {{{\hat{\sigma }}}_{T}}= & 30.17. 
\end{align}</math>
 
 
For rank regression on y:
 
::<math>\begin{align}
  & \widehat{\mu }= & 51.61 \\
& {{{\hat{\sigma }}}_{T}}= & 33.07. 
\end{align}</math>

Latest revision as of 08:44, 3 August 2012

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