Template:Characteristics of the normal distribution - Revision history

Lisa Hacker: Redirected page to The Normal Distribution

2012-08-08T07:44:44Z

Redirected page to The Normal Distribution

← Older revision		Revision as of 07:44, 8 August 2012
Line 1:		Line 1:
	~~==Characteristics of the Normal Distribution==~~		#REDIRECT [[The_Normal_Distribution]]
	~~Some of the specific characteristics of the normal distribution are the following:~~
	:*The normal <math>pdf</math> has a mean, <math>\bar{T}</math> , which is equal to the median, <math>\breve{T}</math>, and also equal to the mode, <math>\tilde{T}</math>, or <math>\bar{T}=\breve{T}=\tilde{T}</math>. This is because the normal distribution is symmetrical about its mean.

	[[~~Image:WB.9 normalpdf.png\|center\|250px\|~~ ]]

	:*The mean, <math>\mu </math> , or the mean life or the <math>MTTF</math> , is also the location parameter of the normal <math>pdf</math> , as it locates the <math>pdf</math> along the abscissa. It can assume values of <math>-\infty <\bar{T}<\infty </math> .
	~~:*The normal <math>pdf</math> has no shape parameter. This means that the normal <math>pdf</math> has only one shape, the bell shape, and this shape does not change.~~

	~~[[Image:WB.9 effect of sigma.png\|center\|250px\| ]]~~

	~~:*The standard deviation, <math>{{\sigma }}</math> , is the scale parameter of the normal <math>pdf</math> .~~

	~~:*As <math>{{\sigma }}</math> decreases, the <math>pdf</math> gets pushed toward the mean, or it becomes narrower and taller.~~

	~~:*As <math>{{\sigma }}</math> increases, the <math>pdf</math> spreads out away from the mean, or it becomes broader and shallower.~~

	~~:*The standard deviation can assume values of <math>0<{{\sigma }}<\infty </math> .~~

	~~:*The greater the variability, the larger the value of <math>{{\sigma }}</math> , and vice versa.~~

	:*The standard deviation is also the distance between the mean and the point of inflection of the <math>pdf</math> , on each side of the mean. The point of inflection is that point of the <math>pdf</math> where the slope changes its value from a decreasing to an increasing one, or where the second derivative of the <math>pdf</math> has a value of zero.

	:*The normal <math>pdf</math> starts at <math>t=-\infty </math> with an <math>f(t)=0</math> . As <math>t</math> increases, <math>f(t)</math> also increases, goes through its point of inflection and reaches its maximum value at <math>t=\bar{T}</math> . Thereafter, <math>f(t)</math> decreases, goes through its point of inflection, and assumes a value of <math>f(t)=0</math> at <math>t=+\infty </math> .


	~~'''Weibull++ Notes on Negative Time Values'''~~

	One of the disadvantages of using the normal distribution for reliability calculations is the fact that the normal distribution starts at negative infinity. This can result in negative values for some of the results. Negative values for time are not accepted in most of the components of Weibull++, nor are they implemented. Certain components of the application reserve negative values for suspensions, or will not return negative results. For example, the Quick Calculation Pad will return a null value (zero) if the result is negative. Only the Free-Form (Probit) data sheet can accept negative values for the random variable (x-axis values).

Nicolette Young: /* Characteristics of the Normal Distribution */

2012-04-25T18:35:52Z

Characteristics of the Normal Distribution

← Older revision		Revision as of 18:35, 25 April 2012
Line 3:		Line 3:
	:*The normal <math>pdf</math> has a mean, <math>\bar{T}</math> , which is equal to the median, <math>\breve{T}</math>, and also equal to the mode, <math>\tilde{T}</math>, or <math>\bar{T}=\breve{T}=\tilde{T}</math>. This is because the normal distribution is symmetrical about its mean.		:*The normal <math>pdf</math> has a mean, <math>\bar{T}</math> , which is equal to the median, <math>\breve{T}</math>, and also equal to the mode, <math>\tilde{T}</math>, or <math>\bar{T}=\breve{T}=\tilde{T}</math>. This is because the normal distribution is symmetrical about its mean.

	[[Image:WB.9 normalpdf.png\|center\|~~400px~~\| ]]		[[Image:WB.9 normalpdf.png\|center\|250px\| ]]

	:*The mean, <math>\mu </math> , or the mean life or the <math>MTTF</math> , is also the location parameter of the normal <math>pdf</math> , as it locates the <math>pdf</math> along the abscissa. It can assume values of <math>-\infty <\bar{T}<\infty </math> .		:*The mean, <math>\mu </math> , or the mean life or the <math>MTTF</math> , is also the location parameter of the normal <math>pdf</math> , as it locates the <math>pdf</math> along the abscissa. It can assume values of <math>-\infty <\bar{T}<\infty </math> .
	:*The normal <math>pdf</math> has no shape parameter. This means that the normal <math>pdf</math> has only one shape, the bell shape, and this shape does not change.		:*The normal <math>pdf</math> has no shape parameter. This means that the normal <math>pdf</math> has only one shape, the bell shape, and this shape does not change.

	[[Image:WB.9 effect of sigma.png\|center\|~~400px~~\| ]]		[[Image:WB.9 effect of sigma.png\|center\|250px\| ]]

	:*The standard deviation, <math>{{\sigma }}</math> , is the scale parameter of the normal <math>pdf</math> .		:*The standard deviation, <math>{{\sigma }}</math> , is the scale parameter of the normal <math>pdf</math> .

Nicolette Young: /* Characteristics of the Normal Distribution */

2012-03-14T19:18:39Z

Characteristics of the Normal Distribution

← Older revision		Revision as of 19:18, 14 March 2012
Line 8:		Line 8:
	:*The normal <math>pdf</math> has no shape parameter. This means that the normal <math>pdf</math> has only one shape, the bell shape, and this shape does not change.		:*The normal <math>pdf</math> has no shape parameter. This means that the normal <math>pdf</math> has only one shape, the bell shape, and this shape does not change.

	[[Image:WB.9 effect of sigma.png~~\|thumb~~\|center\|400px\| ]]		[[Image:WB.9 effect of sigma.png\|center\|400px\| ]]

	:*The standard deviation, <math>{{\sigma }}</math> , is the scale parameter of the normal <math>pdf</math> .		:*The standard deviation, <math>{{\sigma }}</math> , is the scale parameter of the normal <math>pdf</math> .

Nicolette Young: /* Characteristics of the Normal Distribution */

2012-03-14T19:17:39Z

Characteristics of the Normal Distribution

← Older revision		Revision as of 19:17, 14 March 2012
Line 3:		Line 3:
	:*The normal <math>pdf</math> has a mean, <math>\bar{T}</math> , which is equal to the median, <math>\breve{T}</math>, and also equal to the mode, <math>\tilde{T}</math>, or <math>\bar{T}=\breve{T}=\tilde{T}</math>. This is because the normal distribution is symmetrical about its mean.		:*The normal <math>pdf</math> has a mean, <math>\bar{T}</math> , which is equal to the median, <math>\breve{T}</math>, and also equal to the mode, <math>\tilde{T}</math>, or <math>\bar{T}=\breve{T}=\tilde{T}</math>. This is because the normal distribution is symmetrical about its mean.

	[[Image:~~NPDF~~.~~gif\|thumb~~\|center\|~~300px~~\| ]]		[[Image:WB.9 normalpdf.png\|center\|400px\| ]]

	:*The mean, <math>\mu </math> , or the mean life or the <math>MTTF</math> , is also the location parameter of the normal <math>pdf</math> , as it locates the <math>pdf</math> along the abscissa. It can assume values of <math>-\infty <\bar{T}<\infty </math> .		:*The mean, <math>\mu </math> , or the mean life or the <math>MTTF</math> , is also the location parameter of the normal <math>pdf</math> , as it locates the <math>pdf</math> along the abscissa. It can assume values of <math>-\infty <\bar{T}<\infty </math> .
	:*The normal <math>pdf</math> has no shape parameter. This means that the normal <math>pdf</math> has only one shape, the bell shape, and this shape does not change.		:*The normal <math>pdf</math> has no shape parameter. This means that the normal <math>pdf</math> has only one shape, the bell shape, and this shape does not change.

	[[Image:~~EffectofsigmaonNDpdf~~.~~gif~~\|thumb\|center\|~~300px~~\| ]]		[[Image:WB.9 effect of sigma.png\|thumb\|center\|400px\| ]]

	:*The standard deviation, <math>{{\sigma }}</math> , is the scale parameter of the normal <math>pdf</math> .		:*The standard deviation, <math>{{\sigma }}</math> , is the scale parameter of the normal <math>pdf</math> .

Harry Guo: /* Characteristics of the Normal Distribution */

2012-02-10T18:21:34Z

Characteristics of the Normal Distribution

← Older revision		Revision as of 18:21, 10 February 2012
Line 23:		Line 23:

	:*The normal <math>pdf</math> starts at <math>t=-\infty </math> with an <math>f(t)=0</math> . As <math>t</math> increases, <math>f(t)</math> also increases, goes through its point of inflection and reaches its maximum value at <math>t=\bar{T}</math> . Thereafter, <math>f(t)</math> decreases, goes through its point of inflection, and assumes a value of <math>f(t)=0</math> at <math>t=+\infty </math> .		:*The normal <math>pdf</math> starts at <math>t=-\infty </math> with an <math>f(t)=0</math> . As <math>t</math> increases, <math>f(t)</math> also increases, goes through its point of inflection and reaches its maximum value at <math>t=\bar{T}</math> . Thereafter, <math>f(t)</math> decreases, goes through its point of inflection, and assumes a value of <math>f(t)=0</math> at <math>t=+\infty </math> .


	'''Weibull++ Notes on Negative Time Values'''		'''Weibull++ Notes on Negative Time Values'''

	One of the disadvantages of using the normal distribution for reliability calculations is the fact that the normal distribution starts at negative infinity. This can result in negative values for some of the results. Negative values for time are not accepted in most of the components of Weibull++, nor are they implemented. Certain components of the application reserve negative values for suspensions, or will not return negative results. For example, the Quick Calculation Pad will return a null value (zero) if the result is negative. Only the Free-Form (Probit) data sheet can accept negative values for the random variable (x-axis values).		One of the disadvantages of using the normal distribution for reliability calculations is the fact that the normal distribution starts at negative infinity. This can result in negative values for some of the results. Negative values for time are not accepted in most of the components of Weibull++, nor are they implemented. Certain components of the application reserve negative values for suspensions, or will not return negative results. For example, the Quick Calculation Pad will return a null value (zero) if the result is negative. Only the Free-Form (Probit) data sheet can accept negative values for the random variable (x-axis values).

Harry Guo: /* Characteristics of the Normal Distribution */

2012-02-10T18:09:31Z

Characteristics of the Normal Distribution

← Older revision		Revision as of 18:09, 10 February 2012
Line 10:		Line 10:
	[[Image:EffectofsigmaonNDpdf.gif\|thumb\|center\|300px\| ]]		[[Image:EffectofsigmaonNDpdf.gif\|thumb\|center\|300px\| ]]

	:*The standard deviation, <math>{{\sigma ~~}_{T~~}}</math> , is the scale parameter of the normal <math>pdf</math> .		:*The standard deviation, <math>{{\sigma }}</math> , is the scale parameter of the normal <math>pdf</math> .

	:*As <math>{{\sigma ~~}_{T~~}}</math> decreases, the <math>pdf</math> gets pushed toward the mean, or it becomes narrower and taller.		:*As <math>{{\sigma }}</math> decreases, the <math>pdf</math> gets pushed toward the mean, or it becomes narrower and taller.

	:*As <math>{{\sigma ~~}_{T~~}}</math> increases, the <math>pdf</math> spreads out away from the mean, or it becomes broader and shallower.		:*As <math>{{\sigma }}</math> increases, the <math>pdf</math> spreads out away from the mean, or it becomes broader and shallower.

	:*The standard deviation can assume values of <math>0<{{\sigma ~~}_{T~~}}<\infty </math> .		:*The standard deviation can assume values of <math>0<{{\sigma }}<\infty </math> .

	:*The greater the variability, the larger the value of <math>{{\sigma ~~}_{T~~}}</math> , and vice versa.		:*The greater the variability, the larger the value of <math>{{\sigma }}</math> , and vice versa.

	:*The standard deviation is also the distance between the mean and the point of inflection of the <math>pdf</math> , on each side of the mean. The point of inflection is that point of the <math>pdf</math> where the slope changes its value from a decreasing to an increasing one, or where the second derivative of the <math>pdf</math> has a value of zero.		:*The standard deviation is also the distance between the mean and the point of inflection of the <math>pdf</math> , on each side of the mean. The point of inflection is that point of the <math>pdf</math> where the slope changes its value from a decreasing to an increasing one, or where the second derivative of the <math>pdf</math> has a value of zero.

	:*The normal <math>pdf</math> starts at <math>T=-\infty </math> with an <math>f(T)=0</math> . As <math>T</math> increases, <math>f(T)</math> also increases, goes through its point of inflection and reaches its maximum value at <math>T=\bar{T}</math> . Thereafter, <math>f(T)</math> decreases, goes through its point of inflection, and assumes a value of <math>f(T)=0</math> at <math>T=+\infty </math> .		:*The normal <math>pdf</math> starts at <math>t=-\infty </math> with an <math>f(t)=0</math> . As <math>t</math> increases, <math>f(t)</math> also increases, goes through its point of inflection and reaches its maximum value at <math>t=\bar{T}</math> . Thereafter, <math>f(t)</math> decreases, goes through its point of inflection, and assumes a value of <math>f(t)=0</math> at <math>t=+\infty </math> .

	'''Weibull++ Notes on Negative Time Values'''		'''Weibull++ Notes on Negative Time Values'''

	One of the disadvantages of using the normal distribution for reliability calculations is the fact that the normal distribution starts at negative infinity. This can result in negative values for some of the results. Negative values for time are not accepted in most of the components of Weibull++, nor are they implemented. Certain components of the application reserve negative values for suspensions, or will not return negative results. For example, the Quick Calculation Pad will return a null value (zero) if the result is negative. Only the Free-Form (Probit) data sheet can accept negative values for the random variable (x-axis values).		One of the disadvantages of using the normal distribution for reliability calculations is the fact that the normal distribution starts at negative infinity. This can result in negative values for some of the results. Negative values for time are not accepted in most of the components of Weibull++, nor are they implemented. Certain components of the application reserve negative values for suspensions, or will not return negative results. For example, the Quick Calculation Pad will return a null value (zero) if the result is negative. Only the Free-Form (Probit) data sheet can accept negative values for the random variable (x-axis values).

Harry Guo at 18:07, 10 February 2012

2012-02-10T18:07:41Z

Harry Guo at 18:06, 10 February 2012

2012-02-10T18:06:59Z

← Older revision		Revision as of 18:06, 10 February 2012
Line 1:		Line 1:
	==Characteristics of the Normal Distribution==		==Characteristics of the Normal Distribution==
	Some of the specific characteristics of the normal distribution are the following:		Some of the specific characteristics of the normal distribution are the following:
	:• The normal <math>pdf</math> has a mean, <math>\bar{T}</math> , which is equal to the median, <math>\breve{T}</math>, and also equal to the mode, <math>\tilde{T}</math>, or <math>\bar{T}=\breve{T}=\tilde{T}</math>.		: *The normal <math>pdf</math> has a mean, <math>\bar{T}</math> , which is equal to the median, <math>\breve{T}</math>, and also equal to the mode, <math>\tilde{T}</math>, or <math>\bar{T}=\breve{T}=\tilde{T}</math>. This is because the normal distribution is symmetrical about its mean.

	This is because the normal distribution is symmetrical about its mean.

	[[Image:NPDF.gif\|thumb\|center\|300px\| ]]		[[Image:NPDF.gif\|thumb\|center\|300px\| ]]

	:~~•The~~ mean, <math>\mu </math> , or the mean life or the <math>MTTF</math> , is also the location parameter of the normal <math>pdf</math> , as it locates the <math>pdf</math> along the abscissa. It can assume values of <math>-\infty <\bar{T}<\infty </math> .		:*The mean, <math>\mu </math> , or the mean life or the <math>MTTF</math> , is also the location parameter of the normal <math>pdf</math> , as it locates the <math>pdf</math> along the abscissa. It can assume values of <math>-\infty <\bar{T}<\infty </math> .
	:~~•The~~ normal <math>pdf</math> has no shape parameter. This means that the normal <math>pdf</math> has only one shape, the bell shape, and this shape does not change.		:*The normal <math>pdf</math> has no shape parameter. This means that the normal <math>pdf</math> has only one shape, the bell shape, and this shape does not change.

	[[Image:EffectofsigmaonNDpdf.gif\|thumb\|center\|300px\| ]]		[[Image:EffectofsigmaonNDpdf.gif\|thumb\|center\|300px\| ]]

Harry Guo: /* Characteristics of the Normal Distribution */

2012-02-10T18:06:19Z

Characteristics of the Normal Distribution

← Older revision		Revision as of 18:06, 10 February 2012
Line 1:		Line 1:
	==Characteristics of the Normal Distribution==		==Characteristics of the Normal Distribution==
	Some of the specific characteristics of the normal distribution are the following:		Some of the specific characteristics of the normal distribution are the following:
	:• The normal <math>pdf</math> has a mean, <math>\bar{T}</math> ,		:• The normal <math>pdf</math> has a mean, <math>\bar{T}</math> , which is equal to the median, <math>\breve{T}</math>, and also equal to the mode, <math>\tilde{T}</math>, or <math>\bar{T}=\breve{T}=\tilde{T}</math>.
	::which is equal to the median, <math>\breve{T}</math> ,
	~~:::~~and also equal to the mode, <math>\tilde{T}</math> ,
	~~::::~~or <math>\bar{T}=\breve{T}=\tilde{T}</math> .

	This is because the normal distribution is symmetrical about its mean.		This is because the normal distribution is symmetrical about its mean.

Harry Guo: /* Weibull++ Notes on Negative Time Values */

2012-02-10T17:43:35Z

Weibull++ Notes on Negative Time Values

← Older revision		Revision as of 17:43, 10 February 2012
Line 29:		Line 29:
	:• The normal <math>pdf</math> starts at <math>T=-\infty </math> with an <math>f(T)=0</math> . As <math>T</math> increases, <math>f(T)</math> also increases, goes through its point of inflection and reaches its maximum value at <math>T=\bar{T}</math> . Thereafter, <math>f(T)</math> decreases, goes through its point of inflection, and assumes a value of <math>f(T)=0</math> at <math>T=+\infty </math> .		:• The normal <math>pdf</math> starts at <math>T=-\infty </math> with an <math>f(T)=0</math> . As <math>T</math> increases, <math>f(T)</math> also increases, goes through its point of inflection and reaches its maximum value at <math>T=\bar{T}</math> . Thereafter, <math>f(T)</math> decreases, goes through its point of inflection, and assumes a value of <math>f(T)=0</math> at <math>T=+\infty </math> .

	~~===~~Weibull++ Notes on Negative Time Values~~===~~		'''Weibull++ Notes on Negative Time Values'''

	One of the disadvantages of using the normal distribution for reliability calculations is the fact that the normal distribution starts at negative infinity. This can result in negative values for some of the results. Negative values for time are not accepted in most of the components of Weibull++, nor are they implemented. Certain components of the application reserve negative values for suspensions, or will not return negative results. For example, the Quick Calculation Pad will return a null value (zero) if the result is negative. Only the Free-Form (Probit) data sheet can accept negative values for the random variable (x-axis values).		One of the disadvantages of using the normal distribution for reliability calculations is the fact that the normal distribution starts at negative infinity. This can result in negative values for some of the results. Negative values for time are not accepted in most of the components of Weibull++, nor are they implemented. Certain components of the application reserve negative values for suspensions, or will not return negative results. For example, the Quick Calculation Pad will return a null value (zero) if the result is negative. Only the Free-Form (Probit) data sheet can accept negative values for the random variable (x-axis values).

← Older revision		Revision as of 18:07, 10 February 2012
Line 1:		Line 1:
	==Characteristics of the Normal Distribution==		==Characteristics of the Normal Distribution==
	Some of the specific characteristics of the normal distribution are the following:		Some of the specific characteristics of the normal distribution are the following:
	: *The normal <math>pdf</math> has a mean, <math>\bar{T}</math> , which is equal to the median, <math>\breve{T}</math>, and also equal to the mode, <math>\tilde{T}</math>, or <math>\bar{T}=\breve{T}=\tilde{T}</math>. This is because the normal distribution is symmetrical about its mean.		:*The normal <math>pdf</math> has a mean, <math>\bar{T}</math> , which is equal to the median, <math>\breve{T}</math>, and also equal to the mode, <math>\tilde{T}</math>, or <math>\bar{T}=\breve{T}=\tilde{T}</math>. This is because the normal distribution is symmetrical about its mean.

	[[Image:NPDF.gif\|thumb\|center\|300px\| ]]		[[Image:NPDF.gif\|thumb\|center\|300px\| ]]
Line 10:		Line 10:
	[[Image:EffectofsigmaonNDpdf.gif\|thumb\|center\|300px\| ]]		[[Image:EffectofsigmaonNDpdf.gif\|thumb\|center\|300px\| ]]

	:• The standard deviation, <math>{{\sigma }_{T}}</math> , is the scale parameter of the normal <math>pdf</math> .		:*The standard deviation, <math>{{\sigma }_{T}}</math> , is the scale parameter of the normal <math>pdf</math> .

	:• As <math>{{\sigma }_{T}}</math> decreases, the <math>pdf</math> gets pushed toward the mean, or it becomes narrower and taller.		:*As <math>{{\sigma }_{T}}</math> decreases, the <math>pdf</math> gets pushed toward the mean, or it becomes narrower and taller.

	: • As <math>{{\sigma }_{T}}</math> increases, the <math>pdf</math> spreads out away from the mean, or it becomes broader and shallower.		:*As <math>{{\sigma }_{T}}</math> increases, the <math>pdf</math> spreads out away from the mean, or it becomes broader and shallower.

	: • The standard deviation can assume values of <math>0<{{\sigma }_{T}}<\infty </math> .		:*The standard deviation can assume values of <math>0<{{\sigma }_{T}}<\infty </math> .

	: • The greater the variability, the larger the value of <math>{{\sigma }_{T}}</math> , and vice versa.		:*The greater the variability, the larger the value of <math>{{\sigma }_{T}}</math> , and vice versa.

	: • The standard deviation is also the distance between the mean and the point of inflection of the <math>pdf</math> , on each side of the mean. The point of inflection is that point of the <math>pdf</math> where the slope changes its value from a decreasing to an increasing one, or where the second derivative of the <math>pdf</math> has a value of zero.		:*The standard deviation is also the distance between the mean and the point of inflection of the <math>pdf</math> , on each side of the mean. The point of inflection is that point of the <math>pdf</math> where the slope changes its value from a decreasing to an increasing one, or where the second derivative of the <math>pdf</math> has a value of zero.

	:• The normal <math>pdf</math> starts at <math>T=-\infty </math> with an <math>f(T)=0</math> . As <math>T</math> increases, <math>f(T)</math> also increases, goes through its point of inflection and reaches its maximum value at <math>T=\bar{T}</math> . Thereafter, <math>f(T)</math> decreases, goes through its point of inflection, and assumes a value of <math>f(T)=0</math> at <math>T=+\infty </math> .		:*The normal <math>pdf</math> starts at <math>T=-\infty </math> with an <math>f(T)=0</math> . As <math>T</math> increases, <math>f(T)</math> also increases, goes through its point of inflection and reaches its maximum value at <math>T=\bar{T}</math> . Thereafter, <math>f(T)</math> decreases, goes through its point of inflection, and assumes a value of <math>f(T)=0</math> at <math>T=+\infty </math> .

	'''Weibull++ Notes on Negative Time Values'''		'''Weibull++ Notes on Negative Time Values'''

	One of the disadvantages of using the normal distribution for reliability calculations is the fact that the normal distribution starts at negative infinity. This can result in negative values for some of the results. Negative values for time are not accepted in most of the components of Weibull++, nor are they implemented. Certain components of the application reserve negative values for suspensions, or will not return negative results. For example, the Quick Calculation Pad will return a null value (zero) if the result is negative. Only the Free-Form (Probit) data sheet can accept negative values for the random variable (x-axis values).		One of the disadvantages of using the normal distribution for reliability calculations is the fact that the normal distribution starts at negative infinity. This can result in negative values for some of the results. Negative values for time are not accepted in most of the components of Weibull++, nor are they implemented. Certain components of the application reserve negative values for suspensions, or will not return negative results. For example, the Quick Calculation Pad will return a null value (zero) if the result is negative. Only the Free-Form (Probit) data sheet can accept negative values for the random variable (x-axis values).