Life Distributions - Revision history

Lisa Hacker: /* The Gamma Distribution */

2013-02-18T21:49:58Z

The Gamma Distribution

← Older revision		Revision as of 21:49, 18 February 2013
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	=== The Gamma Distribution ===		=== The Gamma Distribution ===
	The gamma distribution is a flexible distribution that may offer a good fit to some sets of life data. Sometimes called the Erlang distribution, the gamma distribution has applications in Bayesian analysis as a prior distribution, and it is also commonly used in ~~queuing~~ theory.		The gamma distribution is a flexible distribution that may offer a good fit to some sets of life data. Sometimes called the Erlang distribution, the gamma distribution has applications in Bayesian analysis as a prior distribution, and it is also commonly used in queueing theory.

	The ''pdf'' of the gamma distribution is given by:		The ''pdf'' of the gamma distribution is given by:

Lisa Hacker: /* The Loglogistic Distribution */

2013-02-18T21:46:32Z

The Loglogistic Distribution

← Older revision		Revision as of 21:46, 18 February 2013
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	=== The Loglogistic Distribution ===		=== The Loglogistic Distribution ===
	As may be ~~summarized~~ from the name, the loglogistic distribution is similar to the logistic distribution. Specifically, the data follows a loglogistic distribution when the natural logarithms of the times-to-failure follow a logistic distribution. Accordingly, the loglogistic and lognormal distributions also share many similarities.		As may be surmised from the name, the loglogistic distribution is similar to the logistic distribution. Specifically, the data follows a loglogistic distribution when the natural logarithms of the times-to-failure follow a logistic distribution. Accordingly, the loglogistic and lognormal distributions also share many similarities.

	The ''pdf'' of the loglogistic distribution is given by:		The ''pdf'' of the loglogistic distribution is given by:

Richard House at 22:10, 24 September 2012

2012-09-24T22:10:48Z

Show changes

Richard House: /* Most Commonly Used Distributions */

2012-09-05T16:46:59Z

Most Commonly Used Distributions

← Older revision		Revision as of 16:46, 5 September 2012
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	== Most Commonly Used Distributions ==		== Most Commonly Used Distributions ==
	There are many different lifetime distributions that can be used to model reliability data. Leemis ~~and others ~~[[Appendix:~~_Weibull_References~~\|[22]]]~~ present~~ a good overview of many of these distributions. In this reference, we will concentrate on the most commonly used and most widely applicable distributions for life data analysis, as outlined in the following sections.		There are many different lifetime distributions that can be used to model reliability data. Leemis [[Appendix:_Life_Data_Analysis_References\|[22]]] presents a good overview of many of these distributions. In this reference, we will concentrate on the most commonly used and most widely applicable distributions for life data analysis, as outlined in the following sections.

	=== The Exponential Distribution ===		=== The Exponential Distribution ===

Richard House at 16:46, 5 September 2012

2012-09-05T16:46:05Z

← Older revision		Revision as of 16:46, 5 September 2012
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	<math>\mu\,\!</math>, the mean, and <math>\sigma\,\!,</math> the standard deviation, are its parameters. Both of these parameters are estimated from the data (i.e., the mean and standard deviation of the data). Once these parameters have been estimated, our function <math>f(t)\,\!</math> is fully defined and we can obtain any value for <math>f(t)\,\!</math> given any value of <math>t\,\!</math>.		<math>\mu\,\!</math>, the mean, and <math>\sigma\,\!,</math> the standard deviation, are its parameters. Both of these parameters are estimated from the data (i.e., the mean and standard deviation of the data). Once these parameters have been estimated, our function <math>f(t)\,\!</math> is fully defined and we can obtain any value for <math>f(t)\,\!</math> given any value of <math>t\,\!</math>.

	Given the mathematical representation of a distribution, we can also derive all of the functions needed for life data analysis, which again will depend only on the value of <math>t\,\!</math> after the value of the distribution parameter or parameters have been estimated from data. For example, we know that the exponential distribution ~~<math>~~pdf~~</math>~~ is given by:		Given the mathematical representation of a distribution, we can also derive all of the functions needed for life data analysis, which again will depend only on the value of <math>t\,\!</math> after the value of the distribution parameter or parameters have been estimated from data. For example, we know that the exponential distribution ''pdf'' is given by:

	::<math>\begin{align}		::<math>\begin{align}
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	\end{align}</math>		\end{align}</math>

	This exact same methodology can be applied to any distribution given its ~~<span class="texhtml">~~''pdf''~~</span>~~, with various degrees of difficulty depending on the complexity of <math>f(t)\,\!</math>.		This exact same methodology can be applied to any distribution given its ''pdf'', with various degrees of difficulty depending on the complexity of <math>f(t)\,\!</math>.

	== Parameter Types ==		== Parameter Types ==
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	'''Shape Parameter'''		'''Shape Parameter'''
	The shape parameter, as the name implies, helps define the shape of a distribution. Some distributions, such as the exponential or normal, do not have a shape parameter since they have a predefined shape that does not change. In the case of the normal distribution, the shape is always the familiar bell shape. The effect of the shape parameter on a distribution is reflected in the shapes of the ~~<math>~~pdf~~</math>~~, the reliability function and the failure rate function.		The shape parameter, as the name implies, helps define the shape of a distribution. Some distributions, such as the exponential or normal, do not have a shape parameter since they have a predefined shape that does not change. In the case of the normal distribution, the shape is always the familiar bell shape. The effect of the shape parameter on a distribution is reflected in the shapes of the ''pdf'', the reliability function and the failure rate function.

	'''Location Parameter'''		'''Location Parameter'''
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	=== The Normal Distribution ===		=== The Normal Distribution ===
	The normal distribution is commonly used for general reliability analysis, times-to-failure of simple electronic and mechanical components, equipment or systems. The ~~<span class="texhtml">~~''pdf''~~</span>~~ of the normal distribution is given by:		The normal distribution is commonly used for general reliability analysis, times-to-failure of simple electronic and mechanical components, equipment or systems. The ''pdf'' of the normal distribution is given by:

	::<math>\begin{align}		::<math>\begin{align}
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	The gamma distribution is a flexible distribution that may offer a good fit to some sets of life data. Sometimes called the Erlang distribution, the gamma distribution has applications in Bayesian analysis as a prior distribution, and it is also commonly used in queuing theory.		The gamma distribution is a flexible distribution that may offer a good fit to some sets of life data. Sometimes called the Erlang distribution, the gamma distribution has applications in Bayesian analysis as a prior distribution, and it is also commonly used in queuing theory.

	The ~~<math>~~pdf~~</math>~~ of the gamma distribution is given by:		The ''pdf'' of the gamma distribution is given by:

	::<math>\begin{align}		::<math>\begin{align}
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	=== The Logistic Distribution ===		=== The Logistic Distribution ===
	The logistic distribution has a shape very similar to the normal distribution (i.e., bell shaped), but with heavier tails. Since the logistic distribution has closed form solutions for the reliability, ~~<math>~~cdf~~</math>~~ and failure rate functions, it is sometimes preferred over the normal distribution, where these functions can only be obtained numerically.		The logistic distribution has a shape very similar to the normal distribution (i.e., bell shaped), but with heavier tails. Since the logistic distribution has closed form solutions for the reliability, ''cdf'' and failure rate functions, it is sometimes preferred over the normal distribution, where these functions can only be obtained numerically.

	The ~~<math>~~pdf~~</math>~~ of the logistic distribution is given by:		The ''pdf'' of the logistic distribution is given by:

	::<math>\begin{align}		::<math>\begin{align}
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	As may be summarized from the name, the loglogistic distribution is similar to the logistic distribution. Specifically, the data follows a loglogistic distribution when the natural logarithms of the times-to-failure follow a logistic distribution. Accordingly, the loglogistic and lognormal distributions also share many similarities.		As may be summarized from the name, the loglogistic distribution is similar to the logistic distribution. Specifically, the data follows a loglogistic distribution when the natural logarithms of the times-to-failure follow a logistic distribution. Accordingly, the loglogistic and lognormal distributions also share many similarities.

	The ~~<math>~~pdf~~</math>~~ of the loglogistic distribution is given by:		The ''pdf'' of the loglogistic distribution is given by:

	::<math> \begin{align}		::<math> \begin{align}
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	The Gumbel distribution is also referred to as the Smallest Extreme Value (SEV) distribution or the Smallest Extreme Value (Type 1) distribution. The Gumbel distribution is appropriate for modeling strength, which is sometimes skewed to the left (e.g., few weak units fail under low stress, while the rest fail at higher stresses). The Gumbel distribution could also be appropriate for modeling the life of products that experience very quick wear out after reaching a certain age.		The Gumbel distribution is also referred to as the Smallest Extreme Value (SEV) distribution or the Smallest Extreme Value (Type 1) distribution. The Gumbel distribution is appropriate for modeling strength, which is sometimes skewed to the left (e.g., few weak units fail under low stress, while the rest fail at higher stresses). The Gumbel distribution could also be appropriate for modeling the life of products that experience very quick wear out after reaching a certain age.

	The ~~<span class="texhtml">~~''pdf''~~</span>~~ of the Gumbel distribution is given by:		The ''pdf'' of the Gumbel distribution is given by:

	::<math>\begin{align}		::<math>\begin{align}

Richard House: /* The Gamma Distribution */

2012-08-28T02:22:42Z

The Gamma Distribution

← Older revision		Revision as of 02:22, 28 August 2012
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	=== The Gamma Distribution ===		=== The Gamma Distribution ===
	The gamma distribution is a flexible distribution that may offer a good fit to some sets of life data. Sometimes called the Erlang distribution, the gamma distribution has applications in Bayesian analysis as a prior distribution, and it is also commonly used in queuing theory. The <math>pdf</math> of the gamma distribution is given by:		The gamma distribution is a flexible distribution that may offer a good fit to some sets of life data. Sometimes called the Erlang distribution, the gamma distribution has applications in Bayesian analysis as a prior distribution, and it is also commonly used in queuing theory.

			The <math>pdf</math> of the gamma distribution is given by:

	::<math>\begin{align}		::<math>\begin{align}

Richard House: /* The Logistic Distribution */

2012-08-28T02:22:24Z

The Logistic Distribution

← Older revision		Revision as of 02:22, 28 August 2012
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	=== The Logistic Distribution ===		=== The Logistic Distribution ===
	The logistic distribution has a shape very similar to the normal distribution (i.e., bell shaped), but with heavier tails. Since the logistic distribution has closed form solutions for the reliability, <~~span class="texhtml"~~>''cdf''</~~span~~> and failure rate functions, it is sometimes preferred over the normal distribution, where these functions can only be obtained numerically. The <~~span class="texhtml"~~>''pdf''</~~span~~> of the logistic distribution is given by:		The logistic distribution has a shape very similar to the normal distribution (i.e., bell shaped), but with heavier tails. Since the logistic distribution has closed form solutions for the reliability, <math>cdf</math> and failure rate functions, it is sometimes preferred over the normal distribution, where these functions can only be obtained numerically.

			The <math>pdf</math> of the logistic distribution is given by:

	::<math>\begin{align}		::<math>\begin{align}
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	\end{align}</math>		\end{align}</math>

	The logistic distribution and its characteristics are presented in [[The Logistic Distribution]].		The logistic distribution and its characteristics are presented in [[The Logistic Distribution]].

	=== The Loglogistic Distribution ===		=== The Loglogistic Distribution ===

Richard House: /* The Loglogistic Distribution */

2012-08-28T02:21:45Z

The Loglogistic Distribution

← Older revision		Revision as of 02:21, 28 August 2012
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	=== The Loglogistic Distribution ===		=== The Loglogistic Distribution ===
	As may be summarized from the name, the loglogistic distribution is similar to the logistic distribution. Specifically, the data follows a loglogistic distribution when the natural logarithms of the times-to-failure follow a logistic distribution. Accordingly, the loglogistic and lognormal distributions also share many similarities. The <math>pdf</math> of the loglogistic distribution is given by:		As may be summarized from the name, the loglogistic distribution is similar to the logistic distribution. Specifically, the data follows a loglogistic distribution when the natural logarithms of the times-to-failure follow a logistic distribution. Accordingly, the loglogistic and lognormal distributions also share many similarities.

			The <math>pdf</math> of the loglogistic distribution is given by:

	::<math> \begin{align}		::<math> \begin{align}

Richard House: /* The Loglogistic Distribution */

2012-08-28T02:21:31Z

The Loglogistic Distribution

← Older revision		Revision as of 02:21, 28 August 2012
Line 162:		Line 162:

	=== The Loglogistic Distribution ===		=== The Loglogistic Distribution ===
	As may be summarized from the name, the loglogistic distribution is similar to the logistic distribution. Specifically, the data follows a loglogistic distribution when the natural logarithms of the times-to-failure follow a logistic distribution. Accordingly, the loglogistic and lognormal distributions also share many similarities. ~~<br>~~The <~~span class="texhtml"~~>''pdf''</~~span~~> of the loglogistic distribution is given by:		As may be summarized from the name, the loglogistic distribution is similar to the logistic distribution. Specifically, the data follows a loglogistic distribution when the natural logarithms of the times-to-failure follow a logistic distribution. Accordingly, the loglogistic and lognormal distributions also share many similarities. The <math>pdf</math> of the loglogistic distribution is given by:

	::<math> \begin{align}		::<math> \begin{align}

Richard House: /* The Gamma Distribution */

2012-08-28T02:20:51Z

The Gamma Distribution

← Older revision		Revision as of 02:20, 28 August 2012
Line 125:		Line 125:

	=== The Gamma Distribution ===		=== The Gamma Distribution ===
	The gamma distribution is a flexible distribution that may offer a good fit to some sets of life data. Sometimes called the Erlang distribution, the gamma distribution has applications in Bayesian analysis as a prior distribution, and it is also commonly used in queuing theory. The <~~span class="texhtml"~~>''pdf ''</~~span~~>of the gamma distribution is given by:		The gamma distribution is a flexible distribution that may offer a good fit to some sets of life data. Sometimes called the Erlang distribution, the gamma distribution has applications in Bayesian analysis as a prior distribution, and it is also commonly used in queuing theory. The <math>pdf</math> of the gamma distribution is given by:

	::<math>\begin{align}		::<math>\begin{align}