Brief Statistical Background - Revision history

Lisa Hacker at 18:39, 18 September 2023

2023-09-18T18:39:33Z

← Older revision		Revision as of 18:39, 18 September 2023
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	<noinclude>{{Navigation box}}[[Category:Shared Articles]]		<noinclude>{{Navigation box}}[[Category:Shared Articles]]
	''This article also appears in the [~~[Basic_Statistical_Background\|~~Life ~~Data Analysis Reference]~~], [~~[Appendix_A~~:~~_Brief_Statistical_Background\|~~Accelerated ~~Life Testing Data Analysis Reference]~~] and [~~[Statistical_Background#A_Brief_Introduction_to_Continuous_Life_Distributions\|~~System ~~Analysis Reference]~~] ~~books~~.'' </noinclude>		''This article also appears in the [https://help.reliasoft.com/reference/life_data_analysis Life data analysis reference], [https://help.reliasoft.com/reference/accelerated_life_testing_data_analysis Accelerated life testing reference] and [https://help.reliasoft.com/reference/system_analysis System analysis reference].'' </noinclude>
	===Random Variables===		===Random Variables===
	<br>		<br>

Sharon Honecker: /* Conditional Reliability Function */ changed R(T,t) to R(t|T)

2017-02-08T20:00:08Z

Conditional Reliability Function: changed R(T,t) to R(t|T)

← Older revision		Revision as of 20:00, 8 February 2017
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	Conditional reliability is the probability of successfully completing another mission following the successful completion of a previous mission. The time of the previous mission and the time for the mission to be undertaken must be taken into account for conditional reliability calculations. The conditional reliability function is given by:		Conditional reliability is the probability of successfully completing another mission following the successful completion of a previous mission. The time of the previous mission and the time for the mission to be undertaken must be taken into account for conditional reliability calculations. The conditional reliability function is given by:
	<br>		<br>
	::<math>R(T,t)=\frac{R(T+t)}{R(T)}\ \,\!</math>		::<math>R(t\|T)=\frac{R(T+t)}{R(T)}\ \,\!</math>

	===Failure Rate Function===		===Failure Rate Function===

Richard House: /* Reliability Function */

2016-01-05T17:11:00Z

Reliability Function

← Older revision		Revision as of 17:11, 5 January 2016
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	The following figure illustrates this.		The following figure illustrates this.

	[[Image:3.6.png\|center\|400px\|Reliability as area under ''pdf''.]]		[[Image:3.6.png\|center\|400px\|Reliability as area under ''pdf''.\|link=]]

	To show this mathematically, we first define the unreliability function, <math>Q(t)\,\!</math>, which is the probability of failure, or the probability that our time-to-failure is in the region of 0 and <math>t\,\!</math>. This is the same as the ''cdf''. So from <math>F(t)=\int_{0}^{t}f(s)ds\ \,\!</math>:		To show this mathematically, we first define the unreliability function, <math>Q(t)\,\!</math>, which is the probability of failure, or the probability that our time-to-failure is in the region of 0 and <math>t\,\!</math>. This is the same as the ''cdf''. So from <math>F(t)=\int_{0}^{t}f(s)ds\ \,\!</math>:

Richard House: /* Mathematical Relationship: pdf and cdf */

2016-01-05T17:10:42Z

Mathematical Relationship: pdf and cdf

← Older revision		Revision as of 17:10, 5 January 2016
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	The ''cdf'' is the area under the probability density function up to a value of <math>x\,\!</math>. The total area under the ''pdf'' is always equal to 1, or mathematically:		The ''cdf'' is the area under the probability density function up to a value of <math>x\,\!</math>. The total area under the ''pdf'' is always equal to 1, or mathematically:

	[[Image:3.5.png\|center\|400px\|Total area under a ''pdf''.]]		[[Image:3.5.png\|center\|400px\|Total area under a ''pdf''.\|link=]]

	::<math>\int_{-\infty}^{+\infty }f(x)dx=1\,\!</math>		::<math>\int_{-\infty}^{+\infty }f(x)dx=1\,\!</math>

Richard House: /* The Probability Density Function and the Cumulative Distribution Function */

2016-01-05T17:10:13Z

The Probability Density Function and the Cumulative Distribution Function

← Older revision		Revision as of 17:10, 5 January 2016
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	The ''pdf'' and ''cdf'' give a complete description of the probability distribution of a random variable. The following figure illustrates a ''pdf''.		The ''pdf'' and ''cdf'' give a complete description of the probability distribution of a random variable. The following figure illustrates a ''pdf''.
	[[Image:3.3.png\|center\|400px\|Example of a ''pdf''.]]		[[Image:3.3.png\|center\|400px\|Example of a ''pdf''.\|link=]]

	The next figures illustrate the ''pdf'' - ''cdf'' relationship.		The next figures illustrate the ''pdf'' - ''cdf'' relationship.
	[[Image:chp3pdf.png\|center\|550px\|Graphical representation of the relationship between ''pdf'' and ''cdf''.]]		[[Image:chp3pdf.png\|center\|550px\|Graphical representation of the relationship between ''pdf'' and ''cdf''.\|link=]]

	If <math>X\,\!</math> is a continuous random variable, then the ''pdf'' of <math>X\,\!</math> is a function, <math>f(x)\,\!</math>, such that for any two numbers, <math>a\,\!</math> and <math>b\,\!</math> with <math>a\le b\,\!</math> :		If <math>X\,\!</math> is a continuous random variable, then the ''pdf'' of <math>X\,\!</math> is a function, <math>f(x)\,\!</math>, such that for any two numbers, <math>a\,\!</math> and <math>b\,\!</math> with <math>a\le b\,\!</math> :

Nicolette Young: /* Mathematical Relationship: pdf and cdf */

2015-12-21T17:25:39Z

Mathematical Relationship: pdf and cdf

← Older revision		Revision as of 17:25, 21 December 2015
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	The ''cdf'' is the area under the probability density function up to a value of <math>x\,\!</math>. The total area under the ''pdf'' is always equal to 1, or mathematically:		The ''cdf'' is the area under the probability density function up to a value of <math>x\,\!</math>. The total area under the ''pdf'' is always equal to 1, or mathematically:

	[[Image:3.5.png\|center\|~~500px~~\|Total area under a ''pdf''.]]		[[Image:3.5.png\|center\|400px\|Total area under a ''pdf''.]]

	::<math>\int_{-\infty}^{+\infty }f(x)dx=1\,\!</math>		::<math>\int_{-\infty}^{+\infty }f(x)dx=1\,\!</math>

Nicolette Young: /* The Probability Density Function and the Cumulative Distribution Function */

2015-12-21T17:25:20Z

The Probability Density Function and the Cumulative Distribution Function

← Older revision		Revision as of 17:25, 21 December 2015
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	The ''pdf'' and ''cdf'' give a complete description of the probability distribution of a random variable. The following figure illustrates a ''pdf''.		The ''pdf'' and ''cdf'' give a complete description of the probability distribution of a random variable. The following figure illustrates a ''pdf''.
	[[Image:3.3.png\|center\|~~300px~~\|Example of a ''pdf''.]]		[[Image:3.3.png\|center\|400px\|Example of a ''pdf''.]]

	The next figures illustrate the ''pdf'' - ''cdf'' relationship.		The next figures illustrate the ''pdf'' - ''cdf'' relationship.
	[[Image:chp3pdf.png\|center\|~~450px~~\|Graphical representation of the relationship between ''pdf'' and ''cdf''.]]		[[Image:chp3pdf.png\|center\|550px\|Graphical representation of the relationship between ''pdf'' and ''cdf''.]]

	If <math>X\,\!</math> is a continuous random variable, then the ''pdf'' of <math>X\,\!</math> is a function, <math>f(x)\,\!</math>, such that for any two numbers, <math>a\,\!</math> and <math>b\,\!</math> with <math>a\le b\,\!</math> :		If <math>X\,\!</math> is a continuous random variable, then the ''pdf'' of <math>X\,\!</math> is a function, <math>f(x)\,\!</math>, such that for any two numbers, <math>a\,\!</math> and <math>b\,\!</math> with <math>a\le b\,\!</math> :
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	The ''cdf'' is the area under the probability density function up to a value of <math>x\,\!</math>. The total area under the ''pdf'' is always equal to 1, or mathematically:		The ''cdf'' is the area under the probability density function up to a value of <math>x\,\!</math>. The total area under the ''pdf'' is always equal to 1, or mathematically:

	[[Image:3.5.png\|center\|~~400px~~\|Total area under a ''pdf''.]]		[[Image:3.5.png\|center\|500px\|Total area under a ''pdf''.]]

	::<math>\int_{-\infty}^{+\infty }f(x)dx=1\,\!</math>		::<math>\int_{-\infty}^{+\infty }f(x)dx=1\,\!</math>

Nicolette Young: /* The Probability Density Function and the Cumulative Distribution Function */

2015-12-21T17:24:25Z

The Probability Density Function and the Cumulative Distribution Function

← Older revision		Revision as of 17:24, 21 December 2015
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	The next figures illustrate the ''pdf'' - ''cdf'' relationship.		The next figures illustrate the ''pdf'' - ''cdf'' relationship.
	[[Image:chp3pdf.png\|center\|~~350px~~\|Graphical representation of the relationship between ''pdf'' and ''cdf''.]]		[[Image:chp3pdf.png\|center\|450px\|Graphical representation of the relationship between ''pdf'' and ''cdf''.]]

	If <math>X\,\!</math> is a continuous random variable, then the ''pdf'' of <math>X\,\!</math> is a function, <math>f(x)\,\!</math>, such that for any two numbers, <math>a\,\!</math> and <math>b\,\!</math> with <math>a\le b\,\!</math> :		If <math>X\,\!</math> is a continuous random variable, then the ''pdf'' of <math>X\,\!</math> is a function, <math>f(x)\,\!</math>, such that for any two numbers, <math>a\,\!</math> and <math>b\,\!</math> with <math>a\le b\,\!</math> :
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	The ''cdf'' is the area under the probability density function up to a value of <math>x\,\!</math>. The total area under the ''pdf'' is always equal to 1, or mathematically:		The ''cdf'' is the area under the probability density function up to a value of <math>x\,\!</math>. The total area under the ''pdf'' is always equal to 1, or mathematically:

	[[Image:3.5.png\|center\|~~300px~~\|Total area under a ''pdf''.]]		[[Image:3.5.png\|center\|400px\|Total area under a ''pdf''.]]

	::<math>\int_{-\infty}^{+\infty }f(x)dx=1\,\!</math>		::<math>\int_{-\infty}^{+\infty }f(x)dx=1\,\!</math>

Richard House: /* Random Variables */

2015-12-11T18:42:52Z

Random Variables

← Older revision		Revision as of 18:42, 11 December 2015
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	===Random Variables===		===Random Variables===
	<br>		<br>
	[[Image:chp3randomvariables.png\|center\|150px\|]]		[[Image:chp3randomvariables.png\|center\|150px\|link=\|]]
	<br>		<br>
	In general, most problems in reliability engineering deal with quantitative measures, such as the time-to-failure of a component, or qualitative measures, such as whether a component is defective or non-defective. We can then use a random variable <math>X\,\!</math> to denote these possible measures.		In general, most problems in reliability engineering deal with quantitative measures, such as the time-to-failure of a component, or qualitative measures, such as whether a component is defective or non-defective. We can then use a random variable <math>X\,\!</math> to denote these possible measures.

Richard House at 20:24, 17 September 2012

2012-09-17T20:24:20Z

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