Template:Exponential Statistical Properties - Revision history

Lisa Hacker: Redirected page to Exponential Distribution Functions

2012-08-10T08:08:21Z

Redirected page to Exponential Distribution Functions

← Older revision		Revision as of 08:08, 10 August 2012
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	==Exponential ~~Statistical Properties==~~		#REDIRECT [[Exponential Distribution Functions]]
	~~===The Mean or MTTF===~~
	~~The mean, <math>\overline{T},</math> or mean time to failure (MTTF) is given by:~~


	~~::<math>\begin{align}~~
	~~\bar{T}= & \int_{\gamma }^{\infty }t\cdot f(t)dt \\~~
	~~= & \int_{\gamma }^{\infty }t\cdot \lambda \cdot {{e}^{-\lambda t}}dt \\~~
	~~= & \gamma +\frac{1}{\lambda }=m~~
	~~\end{align}</math>~~


	Note that when <math>\gamma =0</math>, the MTTF is the inverse of the exponential distribution's constant failure rate. This is only true for the exponential distribution. Most other distributions do not have a constant failure rate. Consequently, the inverse relationship between failure rate and MTTF does not hold for these other distributions.

	~~{{Median}}~~


	~~{{Mode}}~~



	~~{{Standard Deviation}}~~



	~~{{Exponential Reliability Function}}~~

	~~{{One-Parameter Exponential Reliability Function}}~~
	~~{{Exponential Conditional Reliability}}~~



	~~{{Exponential Reliable Life}}~~



	~~{{Exponential Failure Rate Function}}~~

Nicolette Young: /* The Exponential Reliable Life */

2012-01-04T00:16:41Z

The Exponential Reliable Life

← Older revision		Revision as of 00:16, 4 January 2012
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	{{Exponential Reliable Life}}		{{Exponential Reliable Life}}

	~~===The Exponential Reliable Life===~~

	~~The reliable life, or the mission duration for a desired reliability goal, <math>{{t}_{R}}</math>, for the one-parameter exponential distribution is:~~

	~~::<math>R({{t}_{R}})={{e}^{-\lambda ({{t}_{R}}-\gamma )}}</math>~~

	~~::<math>\ln [R({{t}_{R}})]=-\lambda ({{t}_{R}}-\gamma )</math>~~

	~~:or:~~

	~~::<math>{{t}_{R}}=\gamma -\frac{\ln [R({{t}_{R}})]}{\lambda }</math>~~

	{{Exponential Failure Rate Function}}		{{Exponential Failure Rate Function}}

Nicolette Young: /* The Exponential Failure Rate Function */

2012-01-04T00:16:15Z

The Exponential Failure Rate Function

← Older revision		Revision as of 00:16, 4 January 2012
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	{{Exponential Failure Rate Function}}		{{Exponential Failure Rate Function}}
	~~===The Exponential Failure Rate Function===~~
	~~The exponential failure rate function is:~~

	~~::<math>\lambda (T)=\frac{f(T)}{R(T)}=\frac{\lambda {{e}^{-\lambda (T-\gamma )}}}{{{e}^{-\lambda (T-\gamma )}}}=\lambda =\text{constant}</math>~~


	Once again, note that the constant failure rate is a characteristic of the exponential distribution, and special cases of other distributions only. Most other distributions have failure rates that are functions of time.

Nicolette Young: /* The Exponential Conditional Reliability */

2012-01-04T00:15:57Z

The Exponential Conditional Reliability

← Older revision		Revision as of 00:15, 4 January 2012
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	{{Exponential Conditional Reliability}}		{{Exponential Conditional Reliability}}

	~~===The Exponential Conditional Reliability===~~

	The exponential conditional reliability equation gives the reliability for a mission of <math>t</math> duration, having already successfully accumulated <math>T</math> hours of operation up to the start of this new mission. The exponential conditional reliability function is:

			{{Exponential Reliable Life}}

	~~::<math>R(t\|T)=\frac{R(T+t)}{R(T)}=\frac{{{e}^{-\lambda (T+t-\gamma )}}}{{{e}^{-\lambda (T-\gamma )}}}={{e}^{-\lambda t}}</math>~~


	which says that the reliability for a mission of <math>t</math> duration undertaken after the component or equipment has already accumulated <math>T</math> hours of operation from age zero is only a function of the mission duration, and not a function of the age at the beginning of the mission. This is referred to as the memoryless property.

	~~{{Exponential Reliable Life}}~~
	===The Exponential Reliable Life===		===The Exponential Reliable Life===

Nicolette Young: /* One-Parameter Exponential Reliability Function */

2012-01-04T00:15:24Z

One-Parameter Exponential Reliability Function

← Older revision		Revision as of 00:15, 4 January 2012
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	{{One-Parameter Exponential Reliability Function}}		{{One-Parameter Exponential Reliability Function}}
	~~====One-Parameter Exponential Reliability Function====~~

	~~The one-parameter exponential reliability function is given by:~~


	~~::<math>R(T)={{e}^{-\lambda T}}={{e}^{-\tfrac{T}{m}}}</math>~~

	{{Exponential Conditional Reliability}}		{{Exponential Conditional Reliability}}

Nicolette Young: /* The Exponential Reliability Function */

2012-01-04T00:14:58Z

The Exponential Reliability Function

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 {{Exponential Reliability Function}}
 {{One-Parameter Exponential Reliability Function}}
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 {{Exponential Conditional Reliability}}
 ===The Exponential Conditional Reliability===

Nicolette Young: /* The Standard Deviation */

2012-01-04T00:14:33Z

The Standard Deviation

← Older revision		Revision as of 00:14, 4 January 2012
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	{{Standard Deviation}}		{{Standard Deviation}}

	~~===The Standard Deviation===~~
	~~The standard deviation, <math>{{\sigma }_{T}}</math>, is:~~


	~~::<math>~~{{~~\sigma~~ }~~_{T}}=\frac{1}{\lambda~~ }~~=m</math>~~		{{Exponential Reliability Function}}

	~~{{Exponential Reliability Function}}~~
	===The Exponential Reliability Function===		===The Exponential Reliability Function===
	The equation for the two-parameter exponential cumulative density function, or <math>cdf,</math> is given by:		The equation for the two-parameter exponential cumulative density function, or <math>cdf,</math> is given by:

Nicolette Young: /* The Mode */

2012-01-04T00:14:11Z

The Mode

← Older revision		Revision as of 00:14, 4 January 2012
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	{{Mode}}		{{Mode}}

	~~===The Mode===~~
	~~The mode, <math>\tilde{T},</math> is:~~


	~~::<math>\tilde~~{T}~~=\gamma </math>~~		{{Standard Deviation}}

	~~{{Standard Deviation}}~~
	===The Standard Deviation===		===The Standard Deviation===
	The standard deviation, <math>{{\sigma }_{T}}</math>, is:		The standard deviation, <math>{{\sigma }_{T}}</math>, is:

Nicolette Young: /* The Median */

2012-01-04T00:13:49Z

The Median

← Older revision		Revision as of 00:13, 4 January 2012
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	{{Median}}		{{Median}}
	~~===The Median===~~
	~~The median, <math> \breve{T}, </math> is:~~

	~~::<math> \breve{T}=\gamma +\frac{1}{\lambda}\cdot 0.693 </math>~~

	{{Mode}}		{{Mode}}

	===The Mode===		===The Mode===
	The mode, <math>\tilde{T},</math> is:		The mode, <math>\tilde{T},</math> is:

Nicolette Young: Created page with '==Exponential Statistical Properties== ===The Mean or MTTF=== The mean, $\overline{T},$ or mean time to failure (MTTF) is given by: :: $\begin{align} \bar{T}=…'$

2012-01-04T00:13:31Z

Created page with '==Exponential Statistical Properties== ===The Mean or MTTF=== The mean, <math>\overline{T},</math> or mean time to failure (MTTF) is given by: ::<math>\begin{align} \bar{T}=…'

New page

==Exponential Statistical Properties==
===The Mean or MTTF===
The mean, <math>\overline{T},</math> or mean time to failure (MTTF) is given by:

::<math>\begin{align}
\bar{T}= & \int_{\gamma }^{\infty }t\cdot f(t)dt \\
= & \int_{\gamma }^{\infty }t\cdot \lambda \cdot {{e}^{-\lambda t}}dt \\
= & \gamma +\frac{1}{\lambda }=m
\end{align}</math>

Note that when <math>\gamma =0</math>, the MTTF is the inverse of the exponential distribution's constant failure rate. This is only true for the exponential distribution. Most other distributions do not have a constant failure rate. Consequently, the inverse relationship between failure rate and MTTF does not hold for these other distributions.

{{Median}}
===The Median===
The median, <math> \breve{T}, </math> is:

::<math> \breve{T}=\gamma +\frac{1}{\lambda}\cdot 0.693 </math>

{{Mode}}
===The Mode===
The mode, <math>\tilde{T},</math> is:

::<math>\tilde{T}=\gamma </math>

{{Standard Deviation}}
===The Standard Deviation===
The standard deviation, <math>{{\sigma }_{T}}</math>, is:

::<math>{{\sigma }_{T}}=\frac{1}{\lambda }=m</math>

{{Exponential Reliability Function}}
===The Exponential Reliability Function===
The equation for the two-parameter exponential cumulative density function, or <math>cdf,</math> is given by:

::<math>F(T)=Q(T)=1-{{e}^{-\lambda (T-\gamma )}}</math>

Recalling that the reliability function of a distribution is simply one minus the <math>cdf</math>, the reliability function of the two-parameter exponential distribution is given by:

::<math>R(T)=1-Q(T)=1-\int_{0}^{T-\gamma }f(T)dT</math>

::<math>R(T)=1-\int_{0}^{T-\gamma }\lambda {{e}^{-\lambda T}}dT={{e}^{-\lambda (T-\gamma )}}</math>

{{One-Parameter Exponential Reliability Function}}
====One-Parameter Exponential Reliability Function====

The one-parameter exponential reliability function is given by:

::<math>R(T)={{e}^{-\lambda T}}={{e}^{-\tfrac{T}{m}}}</math>

{{Exponential Conditional Reliability}}
===The Exponential Conditional Reliability===

The exponential conditional reliability equation gives the reliability for a mission of <math>t</math> duration, having already successfully accumulated <math>T</math> hours of operation up to the start of this new mission. The exponential conditional reliability function is:

::<math>R(t|T)=\frac{R(T+t)}{R(T)}=\frac{{{e}^{-\lambda (T+t-\gamma )}}}{{{e}^{-\lambda (T-\gamma )}}}={{e}^{-\lambda t}}</math>

which says that the reliability for a mission of <math>t</math> duration undertaken after the component or equipment has already accumulated <math>T</math> hours of operation from age zero is only a function of the mission duration, and not a function of the age at the beginning of the mission. This is referred to as the memoryless property.

{{Exponential Reliable Life}}
===The Exponential Reliable Life===

The reliable life, or the mission duration for a desired reliability goal, <math>{{t}_{R}}</math>, for the one-parameter exponential distribution is:

::<math>R({{t}_{R}})={{e}^{-\lambda ({{t}_{R}}-\gamma )}}</math>

::<math>\ln [R({{t}_{R}})]=-\lambda ({{t}_{R}}-\gamma )</math>

:or:

::<math>{{t}_{R}}=\gamma -\frac{\ln [R({{t}_{R}})]}{\lambda }</math>

{{Exponential Failure Rate Function}}
===The Exponential Failure Rate Function===
The exponential failure rate function is:

::<math>\lambda (T)=\frac{f(T)}{R(T)}=\frac{\lambda {{e}^{-\lambda (T-\gamma )}}}{{{e}^{-\lambda (T-\gamma )}}}=\lambda =\text{constant}</math>

Once again, note that the constant failure rate is a characteristic of the exponential distribution, and special cases of other distributions only. Most other distributions have failure rates that are functions of time.