To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time. 4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Languages
Recent
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds # 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 $X$ whose probability distribution function belongs to such a family, the distribution function of $Y{\stackrel {d}{=}}a+bX$ also belongs to the family (where ${\stackrel {d}{=}}$ means "equal in distribution"—that is, "has the same distribution as"). Moreover, if $X$ and $Y$ are two random variables whose distribution functions are members of the family, and assuming

1. existence of the first two moments and
2. $X$ has zero mean and unit variance,

then $Y$ can be written as $Y{\stackrel {d}{=}}\mu _{Y}+\sigma _{Y}X$ , where $\mu _{Y}$ and $\sigma _{Y}$ are the mean and standard deviation of $Y$ .

In other words, a class $\Omega$ of probability distributions is a location–scale family if for all cumulative distribution functions $F\in \Omega$ and any real numbers $a\in \mathbb {R}$ and $b>0$ , the distribution function $G(x)=F(a+bx)$ is also a member of $\Omega$ .

• If $X$ has a cumulative distribution function $F_{X}(x)=P(X\leq x)$ , then $Y{=}a+bX$ has a cumulative distribution function $F_{Y}(y)=F_{X}\left({\frac {y-a}{b}}\right)$ .
• If $X$ is a discrete random variable with probability mass function $p_{X}(x)=P(X=x)$ , then $Y{=}a+bX$ is a discrete random variable with probability mass function $p_{Y}(y)=p_{X}\left({\frac {y-a}{b}}\right)$ .
• If $X$ is a continuous random variable with probability density function $f_{X}(x)$ , then $Y{=}a+bX$ is a continuous random variable with probability density function $f_{Y}(y)={\frac {1}{b}}f_{X}\left({\frac {y-a}{b}}\right)$ .

In decision theory, if all alternative distributions available to a decision-maker are in the same location–scale family, and the first two moments are finite, then a two-moment decision model can apply, and decision-making can be framed in terms of the means and the variances of the distributions.

## Examples

Often, location–scale families are restricted to those where all members have the same functional form. Most location–scale families are univariate, though not all. Well-known families in which the functional form of the distribution is consistent throughout the family include the following:

## Converting a single distribution to a location–scale family

The following shows how to implement a location–scale family in a statistical package or programming environment where only functions for the "standard" version of a distribution are available. It is designed for R but should generalize to any language and library.

The example here is of the Student's t-distribution, which is normally provided in R only in its standard form, with a single degrees of freedom parameter df. The versions below with _ls appended show how to generalize this to a generalized Student's t-distribution with an arbitrary location parameter mu and scale parameter sigma.

 Probability density function (PDF): dt_ls(x, df, mu, sigma) = 1/sigma * dt((x - mu)/sigma, df) Cumulative distribution function (CDF): pt_ls(x, df, mu, sigma) = pt((x - mu)/sigma, df) Quantile function (inverse CDF): qt_ls(prob, df, mu, sigma) = qt(prob, df)*sigma + mu Generate a random variate: rt_ls(df, mu, sigma) = rt(df)*sigma + mu

Note that the generalized functions do not have standard deviation sigma since the standard t distribution does not have standard deviation of 1.

Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.