Definition:Geometric Distribution/Shifted
Definition
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
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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:
- Expectation of Geometric Distribution: $\expect X = \dfrac p {1 - p}$
- Expectation of Shifted Geometric Distribution: $\expect X = \dfrac 1 p$
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
- Bernoulli Process as Shifted Geometric Distribution where it is shown that this models the number of Bernoulli trials performed before the first success is achieved.
- 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
Sources
- 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $\S 2.2$: Examples: $(9)$
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): geometric distribution
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $13$: Probability distributions
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $15$: Probability distributions