world:ratio-of-uniform
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world:ratio-of-uniform [2023/04/20 08:54] rdouc |
world:ratio-of-uniform [2023/04/20 08:55] rdouc |
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| + | ====== The idea ====== | ||
| The rejection algorithm is based on the following property: | The rejection algorithm is based on the following property: | ||
| * $ (X,Y) \sim \unif\set{(x,y)}{0\leq x\leq M f(y)}$ if and only if $Y \sim f$ and $X|_{Y=y} \sim \unif[0,Mf(y)]$. | * $ (X,Y) \sim \unif\set{(x,y)}{0\leq x\leq M f(y)}$ if and only if $Y \sim f$ and $X|_{Y=y} \sim \unif[0,Mf(y)]$. | ||
| - | * The idea of the ratio-of-uniform method is based on the following property: if $ (U,V) \sim \unif\set{(u,v)}{0\leq u\leq \sqrt{M f(v/u)}}$, then $V/U \sim f$. This can be shown from the change of variable $x=u$, $y=v/u$, i.e. $u=x$, $v=xy$. | + | |
| + | The idea of the ratio-of-uniform method is based on the following property: if $ (U,V) \sim \unif\set{(u,v)}{0\leq u\leq \sqrt{M f(v/u)}}$, then $V/U \sim f$. This can be shown from the change of variable $x=u$, $y=v/u$, i.e. $u=x$, $v=xy$. | ||
| A simple generalisation of this result is: if $ (U,V) \sim \unif\set{(u,v)}{0\leq u\leq G^{-1}\lr{M f\lr{\frac { v} {g(u)}}}}$, then $V/g(U) \sim f$ where $g: \rset^+ \to \rset^+_*$ and $G(x)=\int_0^x g(u) \rmd u$. | A simple generalisation of this result is: if $ (U,V) \sim \unif\set{(u,v)}{0\leq u\leq G^{-1}\lr{M f\lr{\frac { v} {g(u)}}}}$, then $V/g(U) \sim f$ where $g: \rset^+ \to \rset^+_*$ and $G(x)=\int_0^x g(u) \rmd u$. | ||
world/ratio-of-uniform.txt · Last modified: 2023/04/20 08:57 by rdouc
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