Location-scale family

In probability theory, especially in mathematical statistics, a location-scale family is a family of probability distributions parametrized by a location parameter and a non-negative scale parameter. For any random variable ${\displaystyle X}$ whose probability distribution function belongs to such a family, the distribution function of ${\displaystyle Y{\stackrel {d}{=}}a+bX}$ also belongs to the family (where ${\displaystyle {\stackrel {d}{=}}}$ means "equal in distribution"—that is, "has the same distribution as"). Moreover, if ${\displaystyle X}$ and ${\displaystyle Y}$ are two random variables whose distribution functions are members of the family, and ${\displaystyle X}$ has zero mean and unit variance, then ${\displaystyle Y}$ can be written as ${\displaystyle Y{\stackrel {d}{=}}\mu _{Y}+\sigma _{Y}X}$ , where ${\displaystyle \mu _{Y}}$ and ${\displaystyle \sigma _{Y}}$ are the mean and standard deviation of ${\displaystyle Y}$.

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