Location and Scale Families
- Statistics 3.3
in Mathematics on Statistics
·
1 min
Location–Scale Families
A location–scale family is a class of probability distributions that can be expressed in the form:
\[ \frac{1}{\sigma} f\left( \frac{x - \mu}{\sigma} \right) \]
where $\mu$ is the location parameter and $\sigma > 0$ is the scale parameter. The location parameter $\mu$ shifts the distribution along the x-axis, while the scale parameter $\sigma$ stretches or compresses the distribution. Therefore, if the random variable $Z$ has a pdf $f(z)$, then the random variable $X = \sigma Z + \mu$ has a pdf given above. Then we have:
- $\mathrm{E}[X] = \sigma \mathrm{E}[Z] + \mu$
- $\mathrm{Var}[X] = \sigma^2 \mathrm{Var}[Z]$
