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

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:*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.  
:*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> .  
:*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.  
:*The distribution starts at <span class="texhtml">''T'' = 0</span> at the level of <span class="texhtml">''f''(''T'' = 0) = λ</span>. It decreases thereafter exponentially and monotonically as <span class="texhtml">''T''</span> increases, and it is convex.  
:*As <math>T\to \infty </math> , <math>f(T)\to 0</math>.  
:*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>.
:*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>  
<br> [[Image:ALTA4reliabilityvstimeplot.gif|thumb|center|400px]] <br>


==== Effects of <span class="texhtml">λ</span> on the Reliability Function  ====
==== Effects of <span class="texhtml">λ</span> on the Reliability Function  ====

Revision as of 22:46, 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. It decreases thereafter exponentially and monotonically as T increases, and it 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