Appendix A: Generating Random Numbers from a Distribution: Difference between revisions

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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.  
=Generating Random Times from a Weibull Distribution=
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>
<br>
The Weibull reliability function is given by,
<br>
::<math>\begin{align}
R(T)= & 1-F(t) \\
= & {{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}. 
\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:
<br>
::<math>{{T}_{R}}=\eta \cdot {{\left\{ -\ln \left[ {{U}_{R}}[0,1] \right] \right\}}^{\tfrac{1}{\beta }}}.</math>
<br>
==Conditional==
=BlockSim's Random Number Generator (RNG)=
= Sections =
= Sections =
#[[Generating Random Times from a Weibull Distribution]]
#[[Generating Random Times from a Weibull Distribution]]
#[[Conditional]]
#[[Conditional]]
#[[Regarding BlockSim's Random Number Generator (RNG)]]
#[[Regarding BlockSim's Random Number Generator (RNG)]]

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Chapter A3: Appendix A: Generating Random Numbers from a Distribution


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Chapter A3  
Appendix A: Generating Random Numbers from a Distribution  

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

Generating Random Times from a Weibull Distribution

The [math]\displaystyle{ cdf }[/math] of the 2-parameter Weibull distribution is given by,


[math]\displaystyle{ F(T)=1-{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}. }[/math]



The Weibull reliability function is given by,


[math]\displaystyle{ \begin{align} R(T)= & 1-F(t) \\ = & {{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}. \end{align} }[/math]


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


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


Conditional

BlockSim's Random Number Generator (RNG)

Sections

  1. Generating Random Times from a Weibull Distribution
  2. Conditional
  3. Regarding BlockSim's Random Number Generator (RNG)