Probability mass function in the context of "Location parameter"

In statistics, a location parameter of a probability distribution is a scalar- or vector-valued parameter ${\displaystyle x_{0}}$, which determines the "location" or shift of the distribution. In the literature of location parameter estimation, the probability distributions with such parameter are found to be formally defined in one of the following equivalent ways:

• 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: $${\displaystyle g(x'=x-\mu |\sigma )={\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left(-{\frac {1}{2}}\left({\frac {x'}{\sigma }}\right)^{2}\right)}$$thus fulfilling the first of the definitions given above.