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| ===Characteristics===
| | #REDIRECT [[Distributions_Used_in_Accelerated_Testing#The_Lognormal_Distribution]] |
| :• The lognormal distribution is a distribution skewed to the right.
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| :• The <math>pdf</math> starts at zero, increases to its mode, and decreases thereafter.
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| [[Image:chp4pdf.gif|thumb|center|300px|''Pdf'' of the lognormal distribution.]] | |
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| The characteristics of the lognormal distribution can be exemplified by examining the two parameters, the log-mean, <math>({{\overline{T}}^{\prime }}),</math> and the log-std, <math>({{\sigma }_{{{T}'}}}),</math> and the effect they have on the <math>pdf</math> .
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| Looking at the Log-Mean <math>({{\overline{T}}^{\prime }})</math>
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| :• The parameter, <math>\bar{{T}'}</math> , or the log-mean life, or the <math>MTT{F}'</math> in terms of the logarithm of the <math>{T}'s</math> is also the scale parameter, and is a unitless number.
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| :• For the same <math>{{\sigma }_{{{T}'}}}</math> the <math>pdf</math> 's skewness increases as <math>\bar{{T}'}</math> increases.
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| [[Image:chp4pdf2.gif|thumb|center|300px|''Pdf'' of the lognormal distribution with different log-mean values.]]
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| ====Looking at the Log-STD <math>({{\sigma }_{{{T}'}}})</math>====
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| :• The parameter <math>{{\sigma }_{{{T}'}}}</math> , or the standard deviation of the <math>{T}'s</math> in terms of their logarithm or of their <math>{T}'</math> , is also the shape parameter, and not the scale parameter as in the normal <math>pdf</math> . It is a unitless number and assumes only positive values.
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| :• The degree of skewness increases as <math>{{\sigma }_{{{T}'}}}</math> increases, for a given <math>\bar{{T}'}</math> .
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| :• For <math>{{\sigma }_{{{T}'}}}</math> values significantly greater than 1, the <math>pdf</math> rises very sharply in the beginning (i.e. for very small values of <math>T</math> near zero), and essentially follows the ordinate axis, peaks out early, and then decreases sharply like an exponential <math>pdf</math> or a Weibull <math>pdf</math> with <math>0<\beta <1</math> .
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| [[Image:chp4pdf3.gif|thumb|center|300px|''Pdf'' of the lognormal distribution with different log-std values.]]
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