|
|
| (4 intermediate revisions by 3 users not shown) |
| Line 1: |
Line 1: |
| ===Characteristics===
| | #REDIRECT [[Distributions_Used_in_Accelerated_Testing#The_Lognormal_Distribution]] |
| :* The lognormal distribution is a distribution skewed to the right.
| |
| :* The <math>pdf</math> starts at zero, increases to its mode, and decreases thereafter.
| |
| <br>
| |
| [[Image:chp4pdf.png|center|400px|''Pdf'' of the lognormal distribution.]] | |
| <br>
| |
| <br>
| |
| 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>.
| |
| <br>
| |
| '''Looking at the Log-Mean''' <math>({{\overline{T}}^{\prime }})</math>
| |
| :* 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 a unitless number.
| |
| :* For the same <math>{{\sigma }_{{{T}'}}}</math> the <math>pdf</math> 's skewness increases as <math>\bar{{T}'}</math> increases.
| |
| <br>
| |
| <br>
| |
| [[Image:chp4pdf2.png|center|400px|''Pdf'' of the lognormal distribution with different log-mean values.]]
| |
| <br>
| |
| | |
|
| |
| ====Looking at the Log-STD <math>({{\sigma }_{{{T}'}}})</math>====
| |
| :* 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.
| |
| :* The degree of skewness increases as <math>{{\sigma }_{{{T}'}}}</math> increases, for a given <math>\bar{{T}'}</math>.
| |
| :* 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>.
| |
| <br>
| |
| [[Image:chp4pdf3.png|center|400px|''Pdf'' of the lognormal distribution with different log-std values.]]
| |
| <br>
| |