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| ===Statistical Properties Summary===
| | #REDIRECT [[Distributions_Used_in_Accelerated_Testing#The_Lognormal_Distribution]] |
| {{ald mean}}
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| ====The Standard Deviation====
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| :• The standard deviation of the lognormal distribution, <math>{{\sigma }_{T}}</math> , is given by:
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| ::<math>{{\sigma }_{T}}=\sqrt{\left( {{e}^{2\bar{{T}'}+\sigma _{{{T}'}}^{2}}} \right)\left( {{e}^{\sigma _{{{T}'}}^{2}}}-1 \right)}</math>
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| :• The standard deviation of the natural logarithms of the times-to-failure, <math>{{\sigma }_{{{T}'}}}</math> , in terms of <math>\bar{T}</math> and <math>{{\sigma }_{T}}</math> is given by:
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| ::<math>{{\sigma }_{{{T}'}}}=\sqrt{\ln \left( \frac{\sigma _{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right)}</math>
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| <br>
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| ====The Median====
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| :• The median of the lognormal distribution is given by:
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| <br>
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| ::<math>\breve{T}={{e}^{{{\bar{T}}^{\prime }}}}</math>
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| ====The Mode====
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| :• The mode of the lognormal distribution is given by:
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| <br>
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| ::<math>\tilde{T}={{e}^{{{\bar{T}}^{\prime }}-\sigma _{{{T}'}}^{2}}}</math>
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| ====Reliability Function====
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| For the lognormal distribution, the reliability for a mission of time <math>T</math> , starting at age 0, is given by:
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| <br>
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| ::<math>R(T)=\mathop{}_{T}^{\infty }f(t)dt</math>
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| <br>
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| :or:
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| <br>
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| ::<math>R(T)=\mathop{}_{{{T}^{^{\prime }}}}^{\infty }\frac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\overline{{{T}'}}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt</math>
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| <br>
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| There is no closed form solution for the lognormal reliability function. Solutions can be obtained via the use of standard normal tables.
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| <br>
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| <br>
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| ====Lognormal Failure Rate====
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| The lognormal failure rate is given by:
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| <br>
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| ::<math>\lambda (T)=\frac{f(T)}{R(T)}=\frac{\tfrac{1}{{T}'{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{(\tfrac{{T}'-\overline{{{T}'}}}{{{\sigma }_{{{T}'}}}})}^{2}}}}}{\mathop{}_{{{T}'}}^{\infty }\tfrac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{(\tfrac{t-\overline{{{T}'}}}{{{\sigma }_{{{T}'}}}})}^{2}}}}dt}</math>
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