Template:Characteristics of the exponential distribution alta: Difference between revisions

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===Characteristics===
=== 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 characteristics of the 1-parameter exponential distribution can be exemplified by examining its parameter, lambda (<span class="texhtml">λ<span style="font-family: sans-serif;">)</span></span> and the effect lambda has on the <span class="texhtml">''p''''d''''f''</span>, reliability and failure rate functions.<br>  
:• 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> .
==== Effects of <span class="texhtml">λ</span> on the ''pdf''  ====
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[[Image:ALTA4FRvsTP.gif|thumb|center|400px|Failure Rate plot of the exponential distribution.]]  
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[[Image:ALTA4pdf.gif|thumb|center|400px]] <br>
 
:*The scale parameter is <math>\tfrac{1}{\lambda }</math> .
:*As <span class="texhtml">λ</span> is decreased in value, the distribution is stretched out to the right, and as <span class="texhtml">λ</span> 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). Its only parameter is the failure rate, <span class="texhtml">λ</span> .
:*The distribution starts at <span class="texhtml">''T'' = 0</span> at the level of <span class="texhtml">''f''(''T'' = 0) = λ</span>, and it decreases thereafter exponentially and monotonically as <span class="texhtml">''T''</span> increases. The distribution is convex.
:*As <math>T\to \infty </math> , <math>f(T)\to 0</math>.
:*This ''pdf'' can be thought of as a special case of the Weibull&lt;span class="texhtml"&lt;/span&gt; ''pdf'' with <span class="texhtml">β = 1</span>.
 
<br> [[Image:ALTA4reliabilityvstimeplot.gif|thumb|center|400px]] <br>
 
==== Effects of <span class="texhtml">λ</span> on the Reliability Function  ====
 
:*The 1-parameter exponential reliability function starts at the value of 1 at <span class="texhtml">''T'' = 0</span> . It decreases thereafter monotonically and is convex.
:*As <math>T\to \infty </math> , <math>R(T\to \infty )\to 0</math>.
 
==== Effects of <span class="texhtml">λ</span> on the Failure Rate Function  ====
 
The failure rate function for the exponential distribution is constant and it is equal to the parameter <span class="texhtml">λ</span> .  
 
<br> [[Image:ALTA4FRvsTP.gif|thumb|center|400px]] <br>

Revision as of 22:39, 6 March 2012

Characteristics

The characteristics of the 1-parameter exponential distribution can be exemplified by examining its parameter, lambda (λ) and the effect lambda has on the p'd'f, reliability and failure rate functions.

Effects of λ on the pdf


ALTA4pdf.gif


  • The scale parameter is [math]\displaystyle{ \tfrac{1}{\lambda } }[/math] .
  • As λ is decreased in value, the distribution is stretched out to the right, and as λ 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). Its only parameter is the failure rate, λ .
  • The distribution starts at T = 0 at the level of f(T = 0) = λ, and it decreases thereafter exponentially and monotonically as T increases. The distribution is convex.
  • As [math]\displaystyle{ T\to \infty }[/math] , [math]\displaystyle{ f(T)\to 0 }[/math].
  • This pdf can be thought of as a special case of the Weibull<span class="texhtml"</span> pdf with β = 1.


ALTA4reliabilityvstimeplot.gif


Effects of λ on the Reliability Function

  • The 1-parameter exponential reliability function starts at the value of 1 at T = 0 . It decreases thereafter monotonically and is convex.
  • As [math]\displaystyle{ T\to \infty }[/math] , [math]\displaystyle{ R(T\to \infty )\to 0 }[/math].

Effects of λ on the Failure Rate Function

The failure rate function for the exponential distribution is constant and it is equal to the parameter λ .


ALTA4FRvsTP.gif


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