Definition:Geometric Distribution/Shifted

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Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.

$X$ has the shifted geometric distribution with parameter $p$ if and only if:

$\map X \Omega = \set {1, 2, \ldots} = \N_{>0}$
$\map \Pr {X = k} = p \paren {1 - p}^{k-1}$

where $0 < p < 1$.

It is written:

$X \sim \ShiftedGeometric p$

Also presented as

The shifted geometric distribution can also be found presented as:

$\map \Pr {X = k} = q^{k - 1} p$

where $q = 1 - p$.

Also known as

Some sources, for example 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction, refer to this as the geometric distribution, failing adequately to distinguish between this and what is defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ as the geometric distribution.

The distinction may appear subtle, but the two distributions do have subtly different behaviour.

For example (and perhaps most significantly), their expectations are different:

Also, beware confusion: some treatments of this subject define the geometric distribution as the number of failures before the first success, that is:

  • $\map X \Omega = \set {0, 1, 2, \ldots} = \N$
  • $\map \Pr {X = k} = p \paren {1 - p}^k$

which makes this distribution hardly any different from (and therefore, hardly any more useful than) the shifted geometric distribution.

Also see

  • Results about the shifted geometric distribution can be found here.

Technical Note

The $\LaTeX$ code for \(\ShiftedGeometric {p}\) is \ShiftedGeometric {p} .

When the argument is a single character, it is usual to omit the braces:

\ShiftedGeometric p