Location parameter

• either as having a probability density function or probability mass function ${\displaystyle f(x-x_0)}$; or
• having a cumulative distribution function ${\displaystyle F(x-x_0)}$; or
• being defined as resulting from the random variable transformation ${\displaystyle x_0+X}$, where ${\displaystyle X}$ is a random variable with a certain, possibly unknown, distribution. See also § Additive noise.

A direct example of a location parameter is the parameter ${\displaystyle \mu }$ of the normal distribution. To see this, note that the probability density function ${\displaystyle f(x|\mu ,\sigma )}$ of a normal distribution ${\displaystyle \mathcal{N}(\mu ,{\sigma }^2)}$ can have the parameter ${\displaystyle \mu }$ factored out and be written as: $${\displaystyle g(x^{\prime }=x-\mu |\sigma )=\frac{1}{\sigma \sqrt{2\pi }}\exp \left(-\frac{1}{2}{\left(\frac{x^{\prime }}{\sigma }\right)}^2\right)}$$ thus fulfilling the first of the definitions given above.

The above definition indicates, in the one-dimensional case, that if ${\displaystyle x_0}$ is increased, the probability density or mass function shifts rigidly to the right, maintaining its exact shape.

A location parameter can also be found in families having more than one parameter, such as location–scale families. In this case, the probability density function or probability mass function will be a special case of the more general form $${\displaystyle f_{x_0,\theta }(x)=f_{\theta }(x-x_0)}$$ where ${\displaystyle x_0}$ is the location parameter, θ represents additional parameters, and ${\displaystyle f_{\theta }}$ is a function parametrized on the additional parameters.

Definition

Source:

Let ${\displaystyle f(x)}$ be any probability density function and let ${\displaystyle \mu }$ and ${\displaystyle \sigma >0}$ be any given constants. Then the function

$${\displaystyle g(x|\mu ,\sigma )=\frac{1}{\sigma }f\left(\frac{x-\mu }{\sigma }\right)}$$

is a probability density function.

The location family is then defined as follows:

Let ${\displaystyle f(x)}$ be any probability density function. Then the family of probability density functions ${\displaystyle \mathcal{F}=\{f(x-\mu ):\mu \in \mathbb{R}\}}$ is called the location family with standard probability density function ${\displaystyle f(x)}$, where ${\displaystyle \mu }$ is called the location parameter for the family.