Quotients of Uniformly Distributed Random Variables

In many cases random variables having absolutely continuous distributions can be represented as quotients of uniformly distributed random variables.

• Combined with acceptance-rejection sampling (see Section 3.2.3) this yields another type of simulation algorithm.
• The mathematical foundation for this type of algorithm is the following transformation theorem for the density of absolutely continuous random vectors (that has already been mentioned e.g. in Section 1.2.3 of the course ,,Statistik I'').

Theorem 3.8    

• Let ${\mathbf{X}}=(X_1,\ldots,X_n)^\top:\Omega\to\mathbb{R}^n$ be an absolutely continuous random vector with joint density $f_{\mathbf{X}}:\mathbb{R}^n\to[0,\infty)$ and let ${\boldsymbol{\varphi}}=(\varphi_1,\ldots,\varphi_n):\mathbb{R}^n\to\mathbb{R}^n$ be a Borel-measurable function with continuous partial derivatives $\partial\varphi_i/\partial x_j(x_1,\ldots,x_n)$.
• Let now the Borel-set $C\in\mathcal{B}(\mathbb{R}^n)$ be picked in a way such that
  $$\{{\mathbf{x}}\in\mathbb{R}^n:f_{\mathbf{X}}({\mathbf{x}})\not= 0\}\subset C$$
  and
  $$\det\Bigl(\frac{\partial\varphi_i}{\partial x_j}(x_1,\ldots,x_n)\Bigr)\not= 0\,, \qquad\forall {\mathbf{x}}=(x_1,\ldots,x_n)\in C\,,$$
  which ensures that the restriction ${\boldsymbol{\varphi}}:C\to D$ of ${\boldsymbol{\varphi}}$ to the set $C$ is a bijection where $D=\{{\boldsymbol{\varphi}}({\mathbf{x}}): {\mathbf{x}}\in C\}$ denotes the image of $\varphi$.
• Let ${\boldsymbol{\varphi}}^{-1}=(\varphi_1^{-1},\ldots,\varphi_n^{-1}):D\to C$ be the inverse of ${\boldsymbol{\varphi}}:C\to D$.
• Then the random vector ${\mathbf{Y}}={\boldsymbol{\varphi}}({\mathbf{X}})$ is also absolutely continuous and the density $f_{\mathbf{Y}}({\mathbf{y}})$ of ${\mathbf{Y}}$ is given by

  $$f_{\mathbf{Y}}({\mathbf{y}})=\left\{\begin{array}{ll} f_{\mathbf{... ...D$,}\\ [3\jot] 0\,, & \mbox{falls ${\mathbf{y}}\not\in D$.} \end{array}\right.$$ (25)

  
  
  which is the same as

  $$f_{\mathbf{Y}}({\mathbf{y}})=\left\{\begin{array}{ll} f_{\mathbf{... ...D$,}\\ [3\jot] 0\,, & \mbox{falls ${\mathbf{y}}\not\in D$.} \end{array}\right.$$ (26)

  
  

From Theorem 3.8 we obtain the following result concerning the representation of absolutely continuous random variables as quotients of uniformly distributed random variables.

Theorem 3.9    

• Let $f^\prime:\mathbb{R}\to[0,\infty)$ be Borel measurable and bounded such that

  $$0<\int_\mathbb{R}f^\prime(x)\,dx<\infty \qquad\sup_{x\in\mathbb{R}} \vert x\vert\,\sqrt{f^\prime(x)}<\infty\,.$$ (27)

  
  

• Let the random vector $(V_1,V_2)$ be uniformly distributed on the (bounded) Borel set

  $$B=\{(x_1,x_2)\in\mathbb{R}^2:0<x_1<\sqrt{f^\prime(x_2/x_1)}\}\,.$$ (28)

  
  

• Then the quotient $V_2/V_1$ is an absolutely continuous random variable with density $f:\mathbb{R}\to[0,\infty)$ where
  $$f(x)\;=\;\frac{f^\prime(x)}{\int_\mathbb{R}f^\prime(y)\,dy}\;,\qquad\forall\, x\in\mathbb{R}\,.$$

Proof

 

• Notice that (27) implies that the Borel set $B$ defined in (28) is bounded, i.e. $0<\vert B\vert<\infty$. This is due to the following reasons.
  • For $x_2>0$ the inequality $x_1<\sqrt{f^\prime(x_2/x_1)}$ is equivalent to $x_2<x_2/x_1\sqrt{f^\prime(x_2/x_1)}$.
  • If on the other hand $x_2<0$ it is equivalent to $x_2>x_2/x_1\sqrt{f^\prime(x_2/x_1)}$.
  • Therefore

    $$B \subset [0,\sup_{x\in\mathbb{R}} \sqrt{f^\prime(x)}]\times[\inf_{x<0} x\,\sqrt{f^\prime(x)},\, \sup_{x>0} x\,\sqrt{f^\prime(x)}]$$ (29)

    
    
    and
    $$B \subset [0,\sup_{x\in\mathbb{R}} \sqrt{f^\prime(x)}]\times[-\su... ...qrt{f^\prime(x)},\, \sup_{x\in\mathbb{R}} \vert x\vert\,\sqrt{f^\prime(x)}]\,.$$

• The following joint density $f_{(V_1,V_2)}(v_1,v_2)$ of the random vector $(V_1,V_2)$ is thus well defined
  $$f_{(V_1,V_2)}(v_1,v_2)=\vert B\vert^{-1}{1\hspace{-1mm}{\rm I}}\bigl(0<v_1<\sqrt{f^\prime(v_2/v_1)}\bigr)\,.$$

• The function ${\boldsymbol{\varphi}}:C\to C$ where $C=(0,\infty)\times\mathbb{R}$ and $\varphi(x_1,x_2)=(x_1,x_2/x_1)$
  • is a bijection of $C$ onto itself
  • and its functional determinant is given by
    $$\det\,\Bigl(\frac{\partial\varphi_i}{\partial x_j}(x_1,x_2)\Bigr)... ...1}{x_1}\end{array}\Biggr)\;=\;\frac{1}{x_1}\;,\qquad\forall\,(x_1,x_2)\in C\,.$$

• Theorem 3.8 therefore implies
  • that the density $f_{(V_1,V_2/V_1)}(y_1,y_2)$ of the random vector $(V_1,V_2/V_1)$ has the following form:
    $$f_{(V_1,V_2/V_1)}(y_1,y_2)=\vert B\vert^{-1}y_1{1\hspace{-1mm}{\rm I}}\bigl(0<y_1<\sqrt{f^\prime(y_2)}\bigr)$$

  • Moreover, the marginal density $f_{V_2/V_1}(y_2)$ of the second component $V_2/V_1$ von $(V_1,V_2/V_1)$ is given by
    $$f_{V_2/V_1}(y_2)\;=\;\vert B\vert^{-1}\int\limits_0^{\sqrt{f^\prime(y_2)}}y_1\, dy_1 \;=\;\frac{f^\prime(y_2)}{2\,\vert B\vert}\;.$$
    
     
      $\Box$
    
    
    

Example

$\;$ (normal distribution)

• Theorem 3.9 yields a third method to generate N$(0,1)$-distributed pseudo-random numbers (as an alternative to the Box-Muller algorithm from Section 3.2.1 and the polar method explained in Section 3.2.3).
• Consider the function $f^\prime:\mathbb{R}\to[0,\infty)$ where $f^\prime(x)=\exp(-x^2/2)$ for all $x\in\mathbb{R}$. For the bounds in (29) we obtain:
  $$\sup_{x\in\mathbb{R}} \sqrt{f^\prime(x)}=1\,,\qquad \inf_{x<0} x\... ...\prime(x)}=-\sqrt{2/e}\,,\qquad \sup_{x>0} x\,\sqrt{f^\prime(x)}=\sqrt{2/e}\,.$$

• According to Theorem 3.9 a sequence $x_1,x_2,\ldots$ of N$(0,1)$-distributed pseudo-random numbers can now be generated as follows.

  1.

  Generate a $(0,1]$-uniformly distributed pseudo-random number $u$ and a $(-\sqrt{2/e},\,\sqrt{2/e}]$-uniformly distributed pseudo-random number $v$.

  2.

  If $u\ge\sqrt{\exp(-v^2/(2u^2))}$ , i.e., if $\log u\ge -v^2/(4u^2)$ $\Leftrightarrow$ $v^2\ge -4u^2\log u$, then return to step 1.

  3.

  Otherwise put $x=v/u$.