Home page for accesible maths 8 Bivariate Transformations 8.2 Use of Dummy Variables 8.4 Key definitions and Relationships

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8.3 Links Between Standard Distributions

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

Distribution of $X,\ldots$	Transformation	Distribution of $Y$
$X_{i}\sim N(\mu_{i},\sigma^{2}_{i})$ , $i=1,\ldots,n$	$Y=X_{1}+\ldots+X_{n}$	$N(\mu_{1}+\ldots+\mu_{n},\sigma^{2}_{1}+\ldots+\sigma^{2}_{n})$
$X_{i}\sim N(\mu_{i},\sigma^{2}_{i})$ , $i=1,\ldots,n$	$Y=\sum_{i=1}^{n}a_{i}X_{i}$	$N(\sum_{i=1}^{n}a_{i}\mu_{i},\sum_{i=1}^{n}a_{i}^{2}\sigma_{i}^{2})$
$X_{i}\sim\operatorname{Exp}(\beta)$ , $i=1,\ldots,n$	$Y=X_{1}+\ldots+X_{n}$	$\operatorname{Gamma}(n,\beta)$
$X_{i}\sim\operatorname{Gamma}(\alpha_{i},\beta)$ , $i=1,\ldots,n$	$Y=X_{1}+\ldots+X_{n}$	$\operatorname{Gamma}(\alpha_{1}+\ldots+\alpha_{n},\beta)$
$X_{1}\sim\operatorname{Gamma}(\alpha_{i},\beta)$ , $i=1,2$	$Y=\frac{X_{1}}{X_{1}+X_{2}}$	Beta $(\alpha_{1},\alpha_{2})$

Table 8.1: Random variables that are related by transformation. All the variables in the left hand column are assumed to be independent of each other.