Template:Probability Plotting Exponential Distribution - Revision history

Lisa Hacker: Redirected page to The Exponential Distribution#Probability Plotting

2012-08-06T09:10:51Z

Redirected page to The Exponential Distribution#Probability Plotting

← Older revision		Revision as of 09:10, 6 August 2012
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	~~===Probability Plotting===~~		#REDIRECT [[The_Exponential_Distribution#Probability_Plotting]]
	Estimation of the parameters for the exponential distribution via probability plotting is very similar to the process used when dealing with the Weibull distribution. Recall, however, that the appearance of the probability plotting paper and the methods by which the parameters are estimated vary from distribution to distribution, so there will be some noticeable differences. In fact, due to the nature of the exponential <math>cdf</math>, the exponential probability plot is the only one with a negative slope. This is because the y-axis of the exponential probability plotting paper represents the reliability, whereas the y-axis for most of the other life distributions represents the unreliability.

	This is illustrated in the process of linearizing the <math>cdf</math>, which is necessary to construct the exponential probability plotting paper. For the two-parameter exponential distribution the cumulative density function is given by:


	~~::<math>F(t)=1-{{e}^{-\lambda (t-\gamma )}}</math>~~


	~~Taking the natural logarithm of both sides of the above equation yields:~~


	~~::<math>\ln \left~~[ ~~1-F(t) \right]=-\lambda (t-\gamma )</math>~~


	~~or:~~


	~~::<math>\ln~~ [~~1-F(t)~~]~~=\lambda \gamma -\lambda t</math>~~


	~~Now, let:~~


	~~::<math>y=\ln [1-F(t)~~]~~</math>~~


	~~::<math>a=\lambda \gamma </math>~~


	~~and:~~


	~~::<math>b=-\lambda </math>~~


	~~which results in the linear equation of:~~


	~~::<math>y=a+bt</math>~~


	Note that with the exponential probability plotting paper, the y-axis scale is logarithmic and the x-axis scale is linear. This means that the zero value is present only on the x-axis. For <math>t=0</math>, <math>R=1</math> and <math>F(t)=0</math>. So if we were to use <math>F(t)</math> for the y-axis, we would have to plot the point <math>(0,0)</math>. However, since the y-axis is logarithmic, there is no place to plot this on the exponential paper. Also, the failure rate, <math>\lambda </math>, is the negative of the slope of the line, but there is an easier way to determine the value of <math>\lambda </math> from the probability plot, as will be illustrated in the following example.

	~~'''Example 1:'''~~
	~~{{One parameter exponential distribution example Probability Plot}}~~

Harry Guo: /* Probability Plotting */

2012-03-05T16:51:39Z

Probability Plotting

← Older revision		Revision as of 16:51, 5 March 2012
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	Taking the natural logarithm of both sides of ~~Eqn. (Fe)~~ yields:		Taking the natural logarithm of both sides of the above equation yields:

Harry Guo: /* Example 1: */

2012-02-09T17:56:04Z

Example 1:

← Older revision		Revision as of 17:56, 9 February 2012
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	Note that with the exponential probability plotting paper, the y-axis scale is logarithmic and the x-axis scale is linear. This means that the zero value is present only on the x-axis. For <math>t=0</math>, <math>R=1</math> and <math>F(t)=0</math>. So if we were to use <math>F(t)</math> for the y-axis, we would have to plot the point <math>(0,0)</math>. However, since the y-axis is logarithmic, there is no place to plot this on the exponential paper. Also, the failure rate, <math>\lambda </math>, is the negative of the slope of the line, but there is an easier way to determine the value of <math>\lambda </math> from the probability plot, as will be illustrated in the following example.		Note that with the exponential probability plotting paper, the y-axis scale is logarithmic and the x-axis scale is linear. This means that the zero value is present only on the x-axis. For <math>t=0</math>, <math>R=1</math> and <math>F(t)=0</math>. So if we were to use <math>F(t)</math> for the y-axis, we would have to plot the point <math>(0,0)</math>. However, since the y-axis is logarithmic, there is no place to plot this on the exponential paper. Also, the failure rate, <math>\lambda </math>, is the negative of the slope of the line, but there is an easier way to determine the value of <math>\lambda </math> from the probability plot, as will be illustrated in the following example.

	~~====~~Example 1:~~====~~		'''Example 1:'''
	{{One parameter exponential distribution example Probability Plot}}		{{One parameter exponential distribution example Probability Plot}}

Harry Guo at 17:53, 9 February 2012

2012-02-09T17:53:13Z

← Older revision		Revision as of 17:53, 9 February 2012
Line 43:		Line 43:
	Note that with the exponential probability plotting paper, the y-axis scale is logarithmic and the x-axis scale is linear. This means that the zero value is present only on the x-axis. For <math>t=0</math>, <math>R=1</math> and <math>F(t)=0</math>. So if we were to use <math>F(t)</math> for the y-axis, we would have to plot the point <math>(0,0)</math>. However, since the y-axis is logarithmic, there is no place to plot this on the exponential paper. Also, the failure rate, <math>\lambda </math>, is the negative of the slope of the line, but there is an easier way to determine the value of <math>\lambda </math> from the probability plot, as will be illustrated in the following example.		Note that with the exponential probability plotting paper, the y-axis scale is logarithmic and the x-axis scale is linear. This means that the zero value is present only on the x-axis. For <math>t=0</math>, <math>R=1</math> and <math>F(t)=0</math>. So if we were to use <math>F(t)</math> for the y-axis, we would have to plot the point <math>(0,0)</math>. However, since the y-axis is logarithmic, there is no place to plot this on the exponential paper. Also, the failure rate, <math>\lambda </math>, is the negative of the slope of the line, but there is an easier way to determine the value of <math>\lambda </math> from the probability plot, as will be illustrated in the following example.

	Example 1: {{One parameter exponential distribution example Probability Plot}}		====Example 1:====
			{{One parameter exponential distribution example Probability Plot}}

Harry Guo: /* Probability Plotting */

2012-02-09T17:50:40Z

Probability Plotting

← Older revision		Revision as of 17:50, 9 February 2012
Line 43:		Line 43:
	Note that with the exponential probability plotting paper, the y-axis scale is logarithmic and the x-axis scale is linear. This means that the zero value is present only on the x-axis. For <math>t=0</math>, <math>R=1</math> and <math>F(t)=0</math>. So if we were to use <math>F(t)</math> for the y-axis, we would have to plot the point <math>(0,0)</math>. However, since the y-axis is logarithmic, there is no place to plot this on the exponential paper. Also, the failure rate, <math>\lambda </math>, is the negative of the slope of the line, but there is an easier way to determine the value of <math>\lambda </math> from the probability plot, as will be illustrated in the following example.		Note that with the exponential probability plotting paper, the y-axis scale is logarithmic and the x-axis scale is linear. This means that the zero value is present only on the x-axis. For <math>t=0</math>, <math>R=1</math> and <math>F(t)=0</math>. So if we were to use <math>F(t)</math> for the y-axis, we would have to plot the point <math>(0,0)</math>. However, since the y-axis is logarithmic, there is no place to plot this on the exponential paper. Also, the failure rate, <math>\lambda </math>, is the negative of the slope of the line, but there is an easier way to determine the value of <math>\lambda </math> from the probability plot, as will be illustrated in the following example.

	{{One parameter exponential distribution example Probability Plot}}		Example 1: {{One parameter exponential distribution example Probability Plot}}

Harry Guo: /* Probability Plotting */

2012-02-07T23:18:18Z

Probability Plotting

← Older revision		Revision as of 23:18, 7 February 2012
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	::<math>\ln [1-F(T)]=\lambda \gamma -\lambda T</math>		::<math>\ln [1-F(t)]=\lambda \gamma -\lambda t</math>


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	::<math>y=\ln [1-F(T)]</math>		::<math>y=\ln [1-F(t)]</math>


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	::<math>y=a+bT</math>		::<math>y=a+bt</math>

Harry Guo: /* Probability Plotting */

2012-02-07T23:17:23Z

Probability Plotting

← Older revision		Revision as of 23:17, 7 February 2012
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	::<math>F(T)=1-{{e}^{-\lambda (T-\gamma )}}</math>		::<math>F(t)=1-{{e}^{-\lambda (t-\gamma )}}</math>


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	::<math>\ln \left[ 1-F(T) \right]=-\lambda (T-\gamma )</math>		::<math>\ln \left[ 1-F(t) \right]=-\lambda (t-\gamma )</math>

Harry Guo: moved Template:Probability PlottingED to Template:Probability Plotting Exponential Distribution

2012-02-06T23:47:10Z

moved Template:Probability PlottingED to Template:Probability Plotting Exponential Distribution

← Older revision	Revision as of 23:47, 6 February 2012
(No difference)

Harry Guo: /* Probability Plotting */

2012-02-06T23:46:29Z

Probability Plotting

← Older revision		Revision as of 23:46, 6 February 2012
Line 43:		Line 43:
	Note that with the exponential probability plotting paper, the y-axis scale is logarithmic and the x-axis scale is linear. This means that the zero value is present only on the x-axis. For <math>t=0</math>, <math>R=1</math> and <math>F(t)=0</math>. So if we were to use <math>F(t)</math> for the y-axis, we would have to plot the point <math>(0,0)</math>. However, since the y-axis is logarithmic, there is no place to plot this on the exponential paper. Also, the failure rate, <math>\lambda </math>, is the negative of the slope of the line, but there is an easier way to determine the value of <math>\lambda </math> from the probability plot, as will be illustrated in the following example.		Note that with the exponential probability plotting paper, the y-axis scale is logarithmic and the x-axis scale is linear. This means that the zero value is present only on the x-axis. For <math>t=0</math>, <math>R=1</math> and <math>F(t)=0</math>. So if we were to use <math>F(t)</math> for the y-axis, we would have to plot the point <math>(0,0)</math>. However, since the y-axis is logarithmic, there is no place to plot this on the exponential paper. Also, the failure rate, <math>\lambda </math>, is the negative of the slope of the line, but there is an easier way to determine the value of <math>\lambda </math> from the probability plot, as will be illustrated in the following example.

	{{~~one~~ parameter exponential distribution example ~~probabilty plot~~}}		{{One parameter exponential distribution example Probability Plot}}

Harry Guo: /* Probability Plotting */

2012-02-06T23:43:24Z

Probability Plotting

← Older revision		Revision as of 23:43, 6 February 2012
Line 43:		Line 43:
	Note that with the exponential probability plotting paper, the y-axis scale is logarithmic and the x-axis scale is linear. This means that the zero value is present only on the x-axis. For <math>t=0</math>, <math>R=1</math> and <math>F(t)=0</math>. So if we were to use <math>F(t)</math> for the y-axis, we would have to plot the point <math>(0,0)</math>. However, since the y-axis is logarithmic, there is no place to plot this on the exponential paper. Also, the failure rate, <math>\lambda </math>, is the negative of the slope of the line, but there is an easier way to determine the value of <math>\lambda </math> from the probability plot, as will be illustrated in the following example.		Note that with the exponential probability plotting paper, the y-axis scale is logarithmic and the x-axis scale is linear. This means that the zero value is present only on the x-axis. For <math>t=0</math>, <math>R=1</math> and <math>F(t)=0</math>. So if we were to use <math>F(t)</math> for the y-axis, we would have to plot the point <math>(0,0)</math>. However, since the y-axis is logarithmic, there is no place to plot this on the exponential paper. Also, the failure rate, <math>\lambda </math>, is the negative of the slope of the line, but there is an easier way to determine the value of <math>\lambda </math> from the probability plot, as will be illustrated in the following example.

	{{one parameter exponential distribution example}}		{{one parameter exponential distribution example probabilty plot}}