Home page for accesible maths 11 Other useful distributions 11.5 The Extreme Value (aka Gumbel) Distribution A R Commands for Distributions

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11.6 Links Between Standard Distributions

In this section we list further links between standard distributions obtained by univariate and multivariate transformations of independent random variables. Many of these results are important in statistics.

$X$	$g(X)$	$Y$
${\operatorname{\mathsf{Unif}}}(0,1)$	$F_{Y}^{-1}(X)$	$F_{Y}$
$F_{X}$	$F_{X}(X)$	${\operatorname{\mathsf{Unif}}}(0,1)$
${\operatorname{\mathsf{Unif}}}(a,b)$	$(X-a)/(b-a)$	${\operatorname{\mathsf{Unif}}}(0,1)$
${\operatorname{\mathsf{Unif}}}(0,1)$	$a+(b-a)X$	${\operatorname{\mathsf{Unif}}}(a,b)$
$\operatorname{\mathsf{Exp}}(\beta)$	$\beta X$	$\operatorname{\mathsf{Exp}}(1)$
$\operatorname{\mathsf{Exp}}(1)$	$X/\beta$	$\operatorname{\mathsf{Exp}}(\beta)$
$\operatorname{\mathsf{Exp}}(1)$	$X^{1/\alpha}/\beta$	$\operatorname{\mathsf{Weib}}(\alpha,\beta)$
$\operatorname{\mathsf{Weib}}(\alpha,\beta)$	$(\beta X)^{\alpha}$	$\operatorname{\mathsf{Exp}}(1)$
$\operatorname{\mathsf{Gam}}(\alpha,\beta)$	$\beta X$	$\operatorname{\mathsf{Gam}}(\alpha,1)$
$\operatorname{\mathsf{Gam}}(\alpha,1)$	$X/\beta$	$\operatorname{\mathsf{Gam}}(\alpha,\beta)$
$F_{\nu_{1},\nu_{2}}$	$\frac{1}{X}$	$F_{\nu_{2},\nu_{1}}$
$t_{\nu}$	$X^{2}$	$F_{1,\nu}$
$\mbox{Beta}(\alpha_{1},\alpha_{2})$	$\frac{\alpha_{2}}{\alpha_{1}}\frac{X}{1-X}$	$F_{2\alpha_{1},2\alpha_{2}}$
$\operatorname{\mathsf{N}}(\mu,\sigma^{2})$	$(X-\mu)/\sigma$	$\operatorname{\mathsf{N}}(0,1)$
$\operatorname{\mathsf{N}}(0,1)$	$\mu+\sigma X$	$\operatorname{\mathsf{N}}(\mu,\sigma^{2})$
$\operatorname{\mathsf{N}}(0,1)$	$X^{2}$	$\operatorname{\mathsf{Gam}}(1/2,1/2)$
$\operatorname{\mathsf{N}}(\mu,\sigma^{2})$	$\operatorname{\mathsf{Exp}}(X)$	log $\operatorname{\mathsf{N}}(\mu,\sigma^{2})$

Table 11.1: Univariate random variables that are related by transformation.

Distribution of $Z,X,\ldots$	Transformation	Distribution of $Y$
$Z_{i}\sim N(0,1)$ , $i=1,\ldots,n$	$Y=Z_{1}^{2}+\ldots+Z_{n}^{2}$	$\chi^{2}_{n}$
$X_{i}\sim N(\mu_{i},\sigma^{2}_{i})$ , $i=1,\ldots,n$	$Y=X_{1}+\ldots+X_{n}$	$N(\mu_{1}+\ldots+\mu_{n},\sigma^{2}_{1}+\ldots+\sigma^{2}_{n})$
$X_{i}\sim N(\mu_{i},\sigma^{2}_{i})$ , $i=1,\ldots,n$	$Y=\sum_{i=1}^{n}a_{i}X_{i}$	$N(\sum_{i=1}^{n}a_{i}\mu_{i},\sum_{i=1}^{n}a_{i}^{2}\sigma_{i}^{2})$
$Z\sim N(0,1)$ , $X\sim\chi^{2}_{\nu}$	$Y=\frac{Z}{\sqrt{X/\nu}}$	$t_{\nu}$
$X_{1}\sim\chi^{2}_{\nu_{1}}$ , $X_{2}\sim\chi^{2}_{\nu_{2}}$ ,	$Y=\frac{X_{1}/\nu_{1}}{X_{2}/\nu_{2}}$	$F_{\nu_{1},\nu_{2}}$
$X_{i}\sim\mbox{Exp}(\beta)$ , $i=1,\ldots,n$	$Y=X_{1}+\ldots+X_{n}$	$\mbox{Gamma}(n,\beta)$
$X_{i}\sim\mbox{Gamma}(\alpha_{i},\beta)$ , $i=1,\ldots,n$	$Y=X_{1}+\ldots+X_{n}$	$\mbox{Gamma}(\alpha_{1}+\ldots+\alpha_{n},\beta)$
$X_{1}\sim\mbox{Gamma}(\alpha_{i},\beta)$ , $i=1,2$	$Y=\frac{X_{1}}{X_{1}+X_{2}}$	Beta $(\alpha_{1},\alpha_{2})$

Table 11.2: Random variables that are related by transformation from multiple independent RVs. All the variables in the left hand column are assumed independent.