Probability density function - Revision history

DavidB4-bot: /* top */clean up & uniformity

2016-07-13T17:52:36Z

top: clean up & uniformity

RJJensen at 21:10, November 8, 2009

2009-11-08T21:10:28Z

RJJensen at 21:09, November 8, 2009

2009-11-08T21:09:44Z

Aschlafly: condition (3) follows from (1) per talk page, updated accordingly

2009-08-31T12:36:00Z

condition (3) follows from (1) per talk page, updated accordingly

Aschlafly at 03:17, August 31, 2009

2009-08-31T03:17:10Z

Aschlafly: improved the definition; more work needed here

2009-08-31T03:16:31Z

improved the definition; more work needed here

BRichtigen: Adding pics

2008-12-14T14:54:21Z

Adding pics

BRichtigen: there are lots of not continuous density functions

2008-12-06T12:58:51Z

there are lots of not continuous density functions

Qwestor at 15:15, December 13, 2007

2007-12-13T15:15:25Z

Qwestor at 14:59, December 13, 2007

2007-12-13T14:59:13Z

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- {|align="right" border="1"
+{|align="right" border="1"
   |+Two Probability Density Functions
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← Older revision		Revision as of 21:10, November 8, 2009
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−	~~[[Category:probability]]~~
	[[Category:Probability and Statistics]]		[[Category:Probability and Statistics]]

← Older revision		Revision as of 21:09, November 8, 2009
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−	~~[[Category:mathematics]]~~
	[[Category:probability]]		[[Category:probability]]
	[[Category:Probability and Statistics]]		[[Category:Probability and Statistics]]

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 It is necessary to express probability as a density function for a variable or parameter which may take on a continuum of values so that the total probability covering the entire domain of support may converge to a finite value.  The counterpart for a discretely distributed variable is the [[probability mass function]].
-In order to qualify as a ''probability density function'', such a function must meet the following criteria:
+In order to qualify as a ''probability density function'', such a function must satisfy the following two criteria:
 (1) <math> f(x) \geq 0 </math>  <math> \forall x </math>  inside the domain of support.
 (2) <math> \int_{-\infty}^\infty \,f(x)\,dx = 1. </math>  i.e., finitely convergent (to unity by convention).
-(3) <math> \int_{-\infty}^a \,f(x)\,dx \leq \int_{-\infty}^b \,f(x)\,dx </math>  for a<b,  i.e., is non-decreasing
+<math> \int_{-\infty}^a \,f(x)\,dx \leq \int_{-\infty}^b \,f(x)\,dx </math>  for a<b,  i.e., is non-decreasing
 Such a function leads to the definition of an associated [[cumulative distribution function]].

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   |-
   |}
-The '''probability density function''' of a continuous [[random variable]] is the function that provides the likelihood that the variable will have a value in a given interval when the function is integrated over that same interval.  Stated another way, a ''probability density function'' ''f'' is a non-negative valued real function whose value is the probability density of the variable that it is a function of. Since it is a density, the actual probability P that the variable will be in the interval [a,b] is
+The '''probability density function''' of a continuous [[random variable]] is the function that provides the likelihood that the variable will have a value in a given interval when the function is integrated over that same interval.  Stated another way, a '''probability density function''' ''f'' is a non-negative valued real function whose value is the probability density of the variable that it is a function of. Since it is a density, the actual probability P that the variable will be in the interval [a,b] is
 :<math>P(a \leq x \leq b) = \int_a^b f(x) \, dx </math>
 This density function is intended to express mathematically the total apportionment of the values of the variable it represents over the variables entire [[domain]].  This "apportionment" can signify different things in various contexts, such as relative proportion of observations, or [[information]] regarding the residual uncertainty of its true value.
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 [[Category:mathematics]]
 [[Category:probability]]
-[[Category:probability and statistics]]
+[[Category:Probability and Statistics]]

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		+	\|+Two Probability Density Functions
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		+	\|[[Image:Norm.png\|px=133]]
		+	\|-
		+	\|[[Image:Uniform.png\|px=133]]
		+	\|-
		+	\|}
	In [[probability theory]], a ''probability density function'' (say) ''f'' is a non-negative valued real function whose value is the probability density of the variable that it is a function of. Since it is a density, the actual probability P that the variable will be in the interval [a,b] is		In [[probability theory]], a ''probability density function'' (say) ''f'' is a non-negative valued real function whose value is the probability density of the variable that it is a function of. Since it is a density, the actual probability P that the variable will be in the interval [a,b] is

← Older revision		Revision as of 12:58, December 6, 2008
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−	In [[probability theory]], a ''probability density function'' (say) ''f'' is a real ~~valued and continuous~~ function whose value is the probability density of the variable that it is a function of. Since it is a density, the actual probability P that the variable will be in the interval [a,b] is	+	In [[probability theory]], a ''probability density function'' (say) ''f'' is a non-negative valued real function whose value is the probability density of the variable that it is a function of. Since it is a density, the actual probability P that the variable will be in the interval [a,b] is

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-(1) <math> \int_{-\infty}^\infty \,f(x)\,dx = 1. </math>  i.e., finitely convergent (to unity by convention).
+(1) <math> f(x) \geq 0 </math>  <math> \forall x </math>  inside the domain of support.
 (2) <math> \int_{-\infty}^a \,f(x)\,dx \leq \int_{-\infty}^b \,f(x)\,dx </math>  for a<b,  i.e., is non-decreasing
 Such a function leads to the definition of an associated [[cumulative distribution function]].
-If the [[domain]] of the variable is [[finite]], then the infinite limits on the above integrals would be replaced by those bounds, and a third requirement would be:
+If the [[domain]] of the variable is [[finite]], then the infinite limits on the above integrals would be replaced by those bounds, and a fourth requirement would be:
-(3) <math> f(x) = 0 </math>  <math> \forall x </math>  outside the finite domain of support.
+(4) <math> f(x) = 0 </math>  <math> \forall x </math>  outside the finite domain of support.
 [[Category:mathematics]]