Template:Alta exponential distribution: Difference between revisions

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==The Exponential Distribution==
#REDIRECT [[Distributions_Used_in_Accelerated_Testing]]
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The exponential distribution is a very commonly used distribution in reliability engineering. Due to its simplicity, it has been widely employed even in cases to which it does not apply. The exponential distribution is used to describe units that have a constant failure rate.
The single-parameter exponential  <math>pdf</math>  is given by:
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::<math>\begin{align}
  & f(T)= & \lambda {{e}^{-\lambda T}}=\frac{1}{m}{{e}^{-\tfrac{1}{m}T}} \\
& T\ge  & 0,\lambda >0,m>0 
\end{align}</math>
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:where:
::• <math>\lambda =</math> constant failure rate, in failures per unit of measurement, e.g. failures per hour, per cycle, etc.
::• <math>\lambda =\tfrac{1}{m}.</math>
::• <math>m=</math> mean time between failures, or to a failure.
::• <math>T=</math> operating time, life, or age, in hours, cycles, miles, actuations, etc.
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This distribution requires the estimation of only one parameter,  <math>\lambda </math> , for its application.
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{{alta statistical properties summary}}
 
===Characteristics===
The characteristics of the 1-parameter exponential distribution can be exemplified by examining its parameter, lambda,  <math>\lambda ,</math>  and the effect lambda has on the  <math>pdf</math> , reliability and failure rate functions.
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====Effects of  <math>\lambda </math>  on the pdf====
[[Image:ALTA4pdf.gif|thumb|center|400px|''Pdf'' plot of the exponential distribution.]]  
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:• The scale parameter is  <math>\tfrac{1}{\lambda }</math> .
:• As  <math>\lambda </math>  is decreased in value, the distribution is stretched out to the right, and as  <math>\lambda </math>  is increased, the distribution is pushed toward the origin.
:• This distribution has no shape parameter as it has only one shape, i.e. the exponential. The only parameter it has is the failure rate,  <math>\lambda </math> .
:• The distribution starts at  <math>T=0</math>  at the level of  <math>f(T=0)=\lambda </math>  and decreases thereafter exponentially and monotonically as  <math>T</math>  increases, and is convex.
:• As  <math>T\to \infty </math> ,  <math>f(T)\to 0</math>.
:• This  <math>pdf</math>  can be thought of as a special case of the Weibull  <math>pdf</math>  with  <math>\beta =1</math> .
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[[Image:ALTA4reliabilityvstimeplot.gif|thumb|center|400px|Reliability plot of the exponential distribution.]]
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====Effects of  <math>\lambda </math>  on the Reliability Function====
:• The 1-parameter exponential reliability function starts at the value of 1 at  <math>T=0</math> . It decreases thereafter monotonically and is convex.
:• As  <math>T\to \infty </math> ,  <math>R(T\to \infty )\to 0</math>.
Effects of  <math>\lambda </math>  on the Failure Rate Function
 
The failure rate function for the exponential distribution is constant and it is equal to the parameter  <math>\lambda </math> .
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[[Image:ALTA4FRvsTP.gif|thumb|center|400px|Failure Rate plot of the exponential distribution.]]
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Latest revision as of 01:08, 16 August 2012

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