# 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