Appendix A: Generating Random Numbers from a Distribution - Revision history

Lisa Hacker at 19:24, 15 September 2023

2023-09-15T19:24:37Z

← Older revision		Revision as of 19:24, 15 September 2023
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	{{Template:bsbook ~~SUB~~\|Appendix A\|Generating Random Numbers from a Distribution}}		{{Template:bsbook\|Appendix A\|Generating Random Numbers from a Distribution}}
	Simulation involves generating random numbers that belong to a specific distribution. We will illustrate this methodology using the Weibull distribution.		Simulation involves generating random numbers that belong to a specific distribution. We will illustrate this methodology using the Weibull distribution.

Sharon Honecker: /* Conditional */

2017-02-08T21:41:32Z

Conditional

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	::<math>R(t\|T)=\frac{R(T+t)}{R(T)}=\frac{{{e}^{-{{\left( \tfrac{T+t}{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}}\,\!</math>		::<math>R(t\|T)=\frac{R(T+t)}{R(T)}=\frac{{{e}^{-{{\left( \tfrac{T+t}{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}}\,\!</math>

			The random time would be the solution for <math>t\,\!</math> for <math>R(t\|T)={{U}_{R}}[0,1]\,\!</math>.

	=BlockSim's Random Number Generator (RNG)=		=BlockSim's Random Number Generator (RNG)=

Sharon Honecker: /* Generating Random Times from a Weibull Distribution */

2017-02-08T20:25:28Z

Generating Random Times from a Weibull Distribution

← Older revision		Revision as of 20:25, 8 February 2017
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	\end{align}\,\!</math>		\end{align}\,\!</math>

	To generate a random time from a Weibull distribution, with a given <math>\eta \,\!</math> and <math>\beta \,\!</math> a uniform random number from 0 to 1, <math>{{U}_{R}}[0,1]\,\!</math> , is first obtained. The random time from a weibull distribution is then obtained from:		To generate a random time from a Weibull distribution, with a given <math>\eta \,\!</math> and <math>\beta \,\!</math>, a uniform random number from 0 to 1, <math>{{U}_{R}}[0,1]\,\!</math> , is first obtained. The random time from a weibull distribution is then obtained from:

	::<math>{{T}_{R}}=\eta \cdot {{\left\{ -\ln \left[ {{U}_{R}}[0,1] \right] \right\}}^{\tfrac{1}{\beta }}}\,\!</math>		::<math>{{T}_{R}}=\eta \cdot {{\left\{ -\ln \left[ {{U}_{R}}[0,1] \right] \right\}}^{\tfrac{1}{\beta }}}\,\!</math>

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

2017-02-08T20:23:38Z

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

← Older revision		Revision as of 20:23, 8 February 2017
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	The Weibull conditional reliability function is given by:		The Weibull conditional reliability function is given by:

	::<math>R(T,t)=\frac{R(T+t)}{R(T)}=\frac{{{e}^{-{{\left( \tfrac{T+t}{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}}\,\!</math>		::<math>R(t\|T)=\frac{R(T+t)}{R(T)}=\frac{{{e}^{-{{\left( \tfrac{T+t}{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}}\,\!</math>

	=BlockSim's Random Number Generator (RNG)=		=BlockSim's Random Number Generator (RNG)=

Harry Guo: /* References */

2013-11-27T22:51:29Z

References

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	#L'Ecuyer, P., 1988, Communications of the ACM, vol. 31, pp.724-774		#L'Ecuyer, P., 1988, Communications of the ACM, vol. 31, pp.724-774
	#L'Ecuyer, P., 2001, Proceedings of the 2001 Winter Simulation Conference, pp.95-105		#L'Ecuyer, P., 2001, Proceedings of the 2001 Winter Simulation Conference, pp.95-105
	#~~Press,~~ William H., Teukolsky, Saul A., Vetterling, William T., Flannery, Brian R., Numerical		#William H., Teukolsky, Saul A., Vetterling, William T., Flannery, Brian R., Numerical Recipes in C: The Art of Scientific Computing, Second Edition, Cambridge University Press, 1988.
	#Recipes in C: The Art of Scientific Computing, Second Edition, Cambridge University Press, 1988.
	#Peters, Edgar E., Fractal Market Analysis: Applying Chaos Theory to Investment & Economics, John Wiley & Sons, 1994.		#Peters, Edgar E., Fractal Market Analysis: Applying Chaos Theory to Investment & Economics, John Wiley & Sons, 1994.
	#Knuth, Donald E., The Art of Computer Programming: Volume 2 - Seminumerical Algorithms, Third Edition, Addison-Wesley, 1998.		#Knuth, Donald E., The Art of Computer Programming: Volume 2 - Seminumerical Algorithms, Third Edition, Addison-Wesley, 1998.

Richard House at 20:13, 17 September 2012

2012-09-17T20:13:22Z

Richard House: /* Generating Random Times from a Weibull Distribution */

2012-09-04T19:55:38Z

Generating Random Times from a Weibull Distribution

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	=Generating Random Times from a Weibull Distribution=		=Generating Random Times from a Weibull Distribution=

	The ~~<math>~~cdf~~</math>~~ of the 2-parameter Weibull distribution is given by:		The ''cdf'' of the 2-parameter Weibull distribution is given by:

	::<math>F(T)=1-{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}</math>		::<math>F(T)=1-{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}</math>

Lisa Hacker at 06:31, 22 August 2012

2012-08-22T06:31:17Z

← Older revision		Revision as of 06:31, 22 August 2012
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	{{Template:bsbook SUB\|Appendix A\|Generating Random Numbers from a Distribution}}		{{Template:bsbook SUB\|Appendix A\|Generating Random Numbers from a Distribution}}

	Simulation involves generating random numbers that belong to a specific distribution. We will illustrate this methodology using the Weibull distribution.		Simulation involves generating random numbers that belong to a specific distribution. We will illustrate this methodology using the Weibull distribution.

Richard House at 02:04, 10 August 2012

2012-08-10T02:04:16Z

← Older revision		Revision as of 02:04, 10 August 2012
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	\end{align}</math>		\end{align}</math>

	To generate a random time from a Weibull distribution, with a given <math>\eta </math> and <math>\beta </math> a uniform random number from 0 to 1, <math>{{U}_{R}}[0,1]</math> , is first obtained. The random time from a weibull distribution is then obtained from:		To generate a random time from a Weibull distribution, with a given <math>\eta \,\!</math> and <math>\beta \,\!</math> a uniform random number from 0 to 1, <math>{{U}_{R}}[0,1]\,\!</math> , is first obtained. The random time from a weibull distribution is then obtained from:

	::<math>{{T}_{R}}=\eta \cdot {{\left\{ -\ln \left[ {{U}_{R}}[0,1] \right] \right\}}^{\tfrac{1}{\beta }}}</math>		::<math>{{T}_{R}}=\eta \cdot {{\left\{ -\ln \left[ {{U}_{R}}[0,1] \right] \right\}}^{\tfrac{1}{\beta }}}</math>

Richard House: /* Generating Random Times from a Weibull Distribution */

2012-08-07T05:51:34Z

Generating Random Times from a Weibull Distribution

← Older revision		Revision as of 05:51, 7 August 2012
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	The <math>cdf</math> of the 2-parameter Weibull distribution is given by:		The <math>cdf</math> of the 2-parameter Weibull distribution is given by:
	~~<br>~~

	::<math>F(T)=1-{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}</math>		::<math>F(T)=1-{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}</math>
	~~<br>~~

	The Weibull reliability function is given by:		The Weibull reliability function is given by:
	~~<br>~~

	::<math>\begin{align}		::<math>\begin{align}
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	\end{align}</math>		\end{align}</math>

	~~<br>~~
	To generate a random time from a Weibull distribution, with a given <math>\eta </math> and <math>\beta </math> a uniform random number from 0 to 1, <math>{{U}_{R}}[0,1]</math> , is first obtained. The random time from a weibull distribution is then obtained from:		To generate a random time from a Weibull distribution, with a given <math>\eta </math> and <math>\beta </math> a uniform random number from 0 to 1, <math>{{U}_{R}}[0,1]</math> , is first obtained. The random time from a weibull distribution is then obtained from:
	~~<br>~~
	::<math>{{T}_{R}}=\eta \cdot {{\left\{ -\ln \left[ {{U}_{R}}[0,1] \right] \right\}}^{\tfrac{1}{\beta }}}</math>		::<math>{{T}_{R}}=\eta \cdot {{\left\{ -\ln \left[ {{U}_{R}}[0,1] \right] \right\}}^{\tfrac{1}{\beta }}}</math>

	~~<br>~~
	==Conditional==		==Conditional==
	The Weibull conditional reliability function is given by:		The Weibull conditional reliability function is given by:
	~~<br>~~
	::<math>R(T,t)=\frac{R(T+t)}{R(T)}=\frac{{{e}^{-{{\left( \tfrac{T+t}{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}}</math>		::<math>R(T,t)=\frac{R(T+t)}{R(T)}=\frac{{{e}^{-{{\left( \tfrac{T+t}{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}}</math>
	~~<br>~~

	=BlockSim's Random Number Generator (RNG)=		=BlockSim's Random Number Generator (RNG)=

← Older revision		Revision as of 20:13, 17 September 2012
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	The ''cdf'' of the 2-parameter Weibull distribution is given by:		The ''cdf'' of the 2-parameter Weibull distribution is given by:

	::<math>F(T)=1-{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}</math>		::<math>F(T)=1-{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}\,\!</math>

	The Weibull reliability function is given by:		The Weibull reliability function is given by:
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	R(T)= & 1-F(t) \\		R(T)= & 1-F(t) \\
	= & {{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}		= & {{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}
	\end{align}</math>		\end{align}\,\!</math>

	To generate a random time from a Weibull distribution, with a given <math>\eta \,\!</math> and <math>\beta \,\!</math> a uniform random number from 0 to 1, <math>{{U}_{R}}[0,1]\,\!</math> , is first obtained. The random time from a weibull distribution is then obtained from:		To generate a random time from a Weibull distribution, with a given <math>\eta \,\!</math> and <math>\beta \,\!</math> a uniform random number from 0 to 1, <math>{{U}_{R}}[0,1]\,\!</math> , is first obtained. The random time from a weibull distribution is then obtained from:

	::<math>{{T}_{R}}=\eta \cdot {{\left\{ -\ln \left[ {{U}_{R}}[0,1] \right] \right\}}^{\tfrac{1}{\beta }}}</math>		::<math>{{T}_{R}}=\eta \cdot {{\left\{ -\ln \left[ {{U}_{R}}[0,1] \right] \right\}}^{\tfrac{1}{\beta }}}\,\!</math>

	==Conditional==		==Conditional==
	The Weibull conditional reliability function is given by:		The Weibull conditional reliability function is given by:

	::<math>R(T,t)=\frac{R(T+t)}{R(T)}=\frac{{{e}^{-{{\left( \tfrac{T+t}{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}}</math>		::<math>R(T,t)=\frac{R(T+t)}{R(T)}=\frac{{{e}^{-{{\left( \tfrac{T+t}{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}}\,\!</math>

	=BlockSim's Random Number Generator (RNG)=		=BlockSim's Random Number Generator (RNG)=