Font (2 3) - + Letter spacing (4 5) - + Word spacing (6 7) - + Line spacing (8 9) - +

4.6 Links Between Standard Distributions

Below we list some of the relationships that arise from transformations of the standard univariate continuous distributions.

$X$	$g(X)$	$Y$
${\operatorname{\mathsf{Unif}}}(0,1)$	$F_{Y}^{-1}(X)$	$F_{Y}$
$F_{X}$	$F_{X}(X)$	${\operatorname{\mathsf{Unif}}}(0,1)$
${\operatorname{\mathsf{Unif}}}(a,b)$	$(X-a)/(b-a)$	${\operatorname{\mathsf{Unif}}}(0,1)$
${\operatorname{\mathsf{Unif}}}(0,1)$	$a+(b-a)X$	${\operatorname{\mathsf{Unif}}}(a,b)$
$\operatorname{\mathsf{Exp}}(\beta)$	$\beta X$	$\operatorname{\mathsf{Exp}}(1)$
$\operatorname{\mathsf{Exp}}(1)$	$X/\beta$	$\operatorname{\mathsf{Exp}}(\beta)$
$\operatorname{\mathsf{Exp}}(1)$	$X^{1/\alpha}/\beta$	$\operatorname{\mathsf{Weib}}(\alpha,\beta)$
$\operatorname{\mathsf{Weib}}(\alpha,\beta)$	$(\beta X)^{\alpha}$	$\operatorname{\mathsf{Exp}}(1)$
$\operatorname{\mathsf{Gam}}(\alpha,\beta)$	$\beta X$	$\operatorname{\mathsf{Gam}}(\alpha,1)$
$\operatorname{\mathsf{Gam}}(\alpha,1)$	$X/\beta$	$\operatorname{\mathsf{Gam}}(\alpha,\beta)$
$\operatorname{\mathsf{N}}(\mu,\sigma^{2})$	$(X-\mu)/\sigma$	$\operatorname{\mathsf{N}}(0,1)$
$\operatorname{\mathsf{N}}(0,1)$	$\mu+\sigma X$	$\operatorname{\mathsf{N}}(\mu,\sigma^{2})$
$\operatorname{\mathsf{N}}(0,1)$	$X^{2}$	$\operatorname{\mathsf{Gam}}(1/2,1/2)$

Table 4.1: Random variables that are related by transformation.