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world:ratio-of-uniform

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world:ratio-of-uniform [2023/04/20 08:55]
rdouc
world:ratio-of-uniform [2023/04/20 08:57]
rdouc [The idea]
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 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 $G(x)=\int_0^x g(u) \rmd u$. 
  
-As far as I can see, these methods can only be interesting if $V=Y g(X)$ or $U=X$ are easy to simulate when $Y \sim f$ and $X|_Y \sim \unif\lrcb{[0,​f(Y)]}$+As far as I can see, these methods can only be interesting if $V=Y g(X)$ or $U=X$ are easy to simulate when $Y \sim f$ and $X|_Y \sim \unif\lrcb{[0,​f(Y)]}$. This is very linked to rejection algorithm...
  
   * [[https://​dl.acm.org/​doi/​pdf/​10.1145/​355744.355750|ratio-of-uniform method]]   * [[https://​dl.acm.org/​doi/​pdf/​10.1145/​355744.355750|ratio-of-uniform method]]
   * {{ :​mynotes:​bf01889987.pdf | Generalisation of the ratio-of-uniform method}}   * {{ :​mynotes:​bf01889987.pdf | Generalisation of the ratio-of-uniform method}}
world/ratio-of-uniform.txt · Last modified: 2023/04/20 08:57 by rdouc