Template:ProbabilityDensitynCumulativeDistributionFunctions - Revision history

Lisa Hacker: Redirected page to Basic Statistical Background

2012-08-09T10:08:21Z

Redirected page to Basic Statistical Background

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	[[~~Category: For Deletion~~]]		#REDIRECT [[Basic_Statistical_Background]]

Lisa Hacker: Replaced content with 'Category: For Deletion'

2012-07-20T12:08:09Z

Replaced content with 'Category: For Deletion'

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	~~==='''The Probability Density and Cumulative Distribution Functions'''===~~		[[Category: For Deletion]]
	~~'''Designations'''~~

	~~From probability and statistics, given a continuous random variable, we denote:~~

	~~:*The probability density function, ''pdf'', as ''f(x)''.~~
	~~:*The cumulative distribution function, ''cdf'', as ''F(x)''.~~

	~~The ''pdf'' and ''cdf'' give a complete description of the probability distribution of a random variable.~~

	~~'''Definitions'''~~

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

	~~::<math>P(a \le X \le b)=\int_a^b f(x)dx</math> and <math>f(x)\ge 0 </math> for all x.~~

	~~That is, the probability that takes on a value in the interval [a,b] is the area under the density function from <math>a</math> to <math>b</math>.~~
	~~The cumulative distribution function, <math>cdf</math>, is a function <math>F(x)</math> of a random variable, <math>X</math>, and is defined for a number <math>x</math> by:~~

	~~[[Image:pdf_cdf.png\|center\|200px\|]]~~

	~~::<math>F(x)=P(X\le x)=\int_0^x f(s)ds </math>~~

	~~That is, for a given value <math>x</math>, <math>F(x)</math> is the probability that the observed value of <math>X</math> will be at most <math>x</math>.~~
	Note that the limits of integration depend on the domain of <math>f(x)</math>. For example, for all the distributions considered in this reference, this domain would be <math>[0,+\infty]</math>, <math>[-\infty ,+\infty]</math> or <math>[\gamma ,+\infty]</math>. In the case of <math>[\gamma ,+\infty ]</math>, we use the constant <math>\gamma </math> to denote an arbitrary non-zero point (or a location that indicates the starting point for the distribution). The next figure illustrates the relationship between the probability density function and the cumulative distribution function.
	~~<br>~~
	~~<br>~~
	~~[[Image:pdf_cdf2.png\|200px\|center]]~~

	~~'''Mathematical Relationship Between the <math>pdf</math> and <math>cdf</math>'''~~

	~~The mathematical relationship between the <math>pdf</math> and <math>cdf</math> is given by:~~

	~~::<math>F(x)=\int_{-\infty }^x f(s)ds</math>~~

	~~Conversely:~~

	~~::<math>f(x)=\frac{d(F(x))}{dx}</math>~~

	In plain English, the value of the <math>cdf</math> at <math>x</math> is the area under the probability density function up to <math>x</math>, if so chosen. It should also be pointed out that the '''total area under the ''' <math>pdf</math> '''is always equal to 1''', or mathematically:

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


	~~[[Image:100_percent.png\|center\|200px\|]]~~


	~~An example of a probability density function is the well-known normal distribution, whose <math>pdf</math> is given by:~~

	~~::<math>f(t)={\frac{1}{\sigma \sqrt{2\pi }}}{e^{-\frac{1}{2}}(\frac{t-\mu}{\sigma})^2}</math>~~

	where <math>\mu </math> is the mean and <math>\sigma</math> is the standard deviation. The normal distribution is a ''two-parameter distribution'', ''i.e.'' with two parameters <math>\mu </math> and <math>\sigma </math>.
	~~Another two-parameter distribution is the lognormal distribution, whose <math>pdf</math> is given by:~~

	~~::<math>f(t)=\frac{1}{t\cdot {{\sigma }^{\prime }}\sqrt{2\pi }}{e}^{-\tfrac{1}{2}(\tfrac{t^{\prime}-{\mu^{\prime}}}{\sigma^{\prime}})^2}</math>~~

	where <math> t^{\prime}</math> is the natural logarithm of the times-to-failure, <math>\mu^{\prime}</math> is the mean of the natural logarithms of the times-to-failure and <math>\sigma^{\prime}</math> is the standard deviation of the natural logarithms of the times-to-failure, <math> t^{\prime }</math>.

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Lisa Hacker at 07:47, 18 July 2012

2012-07-18T07:47:15Z

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	where <math> t^{\prime}</math> is the natural logarithm of the times-to-failure, <math>\mu^{\prime}</math> is the mean of the natural logarithms of the times-to-failure and <math>\sigma^{\prime}</math> is the standard deviation of the natural logarithms of the times-to-failure, <math> t^{\prime }</math>.		where <math> t^{\prime}</math> is the natural logarithm of the times-to-failure, <math>\mu^{\prime}</math> is the mean of the natural logarithms of the times-to-failure and <math>\sigma^{\prime}</math> is the standard deviation of the natural logarithms of the times-to-failure, <math> t^{\prime }</math>.

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Lisa Hacker at 07:45, 18 July 2012

2012-07-18T07:45:54Z

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	where <math> t^{\prime}</math> is the natural logarithm of the times-to-failure, <math>\mu^{\prime}</math> is the mean of the natural logarithms of the times-to-failure and <math>\sigma^{\prime}</math> is the standard deviation of the natural logarithms of the times-to-failure, <math> t^{\prime }</math>.		where <math> t^{\prime}</math> is the natural logarithm of the times-to-failure, <math>\mu^{\prime}</math> is the mean of the natural logarithms of the times-to-failure and <math>\sigma^{\prime}</math> is the standard deviation of the natural logarithms of the times-to-failure, <math> t^{\prime }</math>.

	~~{{Category:For Deletion}}~~

Lisa Hacker at 07:45, 18 July 2012

2012-07-18T07:45:41Z

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	where <math> t^{\prime}</math> is the natural logarithm of the times-to-failure, <math>\mu^{\prime}</math> is the mean of the natural logarithms of the times-to-failure and <math>\sigma^{\prime}</math> is the standard deviation of the natural logarithms of the times-to-failure, <math> t^{\prime }</math>.		where <math> t^{\prime}</math> is the natural logarithm of the times-to-failure, <math>\mu^{\prime}</math> is the mean of the natural logarithms of the times-to-failure and <math>\sigma^{\prime}</math> is the standard deviation of the natural logarithms of the times-to-failure, <math> t^{\prime }</math>.

			{{Category:For Deletion}}

Nicolette Young: /* The Probability Density and Cumulative Distribution Functions */

2012-04-25T16:43:03Z

The Probability Density and Cumulative Distribution Functions

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	The cumulative distribution function, <math>cdf</math>, is a function <math>F(x)</math> of a random variable, <math>X</math>, and is defined for a number <math>x</math> by:		The cumulative distribution function, <math>cdf</math>, is a function <math>F(x)</math> of a random variable, <math>X</math>, and is defined for a number <math>x</math> by:

	[[Image:pdf_cdf.png\|center\|~~400px~~\|]]		[[Image:pdf_cdf.png\|center\|200px\|]]

	::<math>F(x)=P(X\le x)=\int_0^x f(s)ds </math>		::<math>F(x)=P(X\le x)=\int_0^x f(s)ds </math>
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	<br>		<br>
	<br>		<br>
	[[Image:pdf_cdf2.png\|~~400px~~\|center]]		[[Image:pdf_cdf2.png\|200px\|center]]

	'''Mathematical Relationship Between the <math>pdf</math> and <math>cdf</math>'''		'''Mathematical Relationship Between the <math>pdf</math> and <math>cdf</math>'''
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	[[Image:100_percent.png\|center\|~~400px~~\|]]		[[Image:100_percent.png\|center\|200px\|]]

Nicolette Young: /* The Probability Density and Cumulative Distribution Functions */

2012-03-12T16:25:15Z

The Probability Density and Cumulative Distribution Functions

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	[[Image:100_percent.png~~\|thumb~~\|center\|400px\|]]		[[Image:100_percent.png\|center\|400px\|]]

Nicolette Young: /* The Probability Density and Cumulative Distribution Functions */

2012-03-12T16:22:21Z

The Probability Density and Cumulative Distribution Functions

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	The cumulative distribution function, <math>cdf</math>, is a function <math>F(x)</math> of a random variable, <math>X</math>, and is defined for a number <math>x</math> by:		The cumulative distribution function, <math>cdf</math>, is a function <math>F(x)</math> of a random variable, <math>X</math>, and is defined for a number <math>x</math> by:

	[[Image:~~yellow~~.png~~\|thumb~~\|center\|400px\|]]		[[Image:pdf_cdf.png\|center\|400px\|]]

	::<math>F(x)=P(X\le x)=\int_0^x f(s)ds </math>		::<math>F(x)=P(X\le x)=\int_0^x f(s)ds </math>
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	<br>		<br>
	<br>		<br>
	[[Image:~~f2a2p~~.png\|~~thumb~~\|center~~\|400px\|A graphical representation of the relationship between the ''pdf'' and ''cdf''.~~]]		[[Image:pdf_cdf2.png\|400px\|center]]

	'''Mathematical Relationship Between the <math>pdf</math> and <math>cdf</math>'''		'''Mathematical Relationship Between the <math>pdf</math> and <math>cdf</math>'''
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	[[Image:~~area-1~~.png\|thumb\|center\|400px\|]]		[[Image:100_percent.png\|thumb\|center\|400px\|]]

Harry Guo: /* The Probability Density and Cumulative Distribution Functions */

2012-02-09T23:25:18Z

The Probability Density and Cumulative Distribution Functions

← Older revision		Revision as of 23:25, 9 February 2012
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	'''Mathematical Relationship Between the <math>pdf</math> and <math>cdf</math>'''		'''Mathematical Relationship Between the <math>pdf</math> and <math>cdf</math>'''

	The mathematical relationship between the <math>pdf</math> and <math>cdf</math> is given by:		The mathematical relationship between the <math>pdf</math> and <math>cdf</math> is given by:

Harry Guo: /* Mathematical Relationship Between the pdf and cdf */

2012-02-09T23:24:53Z

Mathematical Relationship Between the pdf and cdf

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	[[Image:f2a2p.png\|thumb\|center\|400px\|A graphical representation of the relationship between the ''pdf'' and ''cdf''.]]		[[Image:f2a2p.png\|thumb\|center\|400px\|A graphical representation of the relationship between the ''pdf'' and ''cdf''.]]

	~~====~~Mathematical Relationship Between the <math>pdf</math> and <math>cdf</math>~~====~~		'''Mathematical Relationship Between the <math>pdf</math> and <math>cdf</math>'''
	The mathematical relationship between the <math>pdf</math> and <math>cdf</math> is given by:		The mathematical relationship between the <math>pdf</math> and <math>cdf</math> is given by: