Definition:Poisson Distribution

Definition

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

Then $X$ has the Poisson distribution with parameter $\lambda$ (where $\lambda > 0$) if and only if:

$\Img X = \set {0, 1, 2, \ldots} = \N$

$\map \Pr {X = k} = \dfrac 1 {k!} \lambda^k e^{-\lambda}$

It is written:

$X \sim \Poisson \lambda$

Also denoted as

Some sources denote the Poisson distribution as:

$X \sim \map {\operatorname {Pois} } \lambda$

Some sources use $\mu$ instead of $\lambda$, but this can cause confusion with instances where $\mu$ is used for the expectation.

However, as the expectation of $\Poisson \lambda$ is also $\lambda$, this may not be as much of a confusion as all that.

Also see

• Poisson Distribution Gives Rise to Probability Mass Function satisfying $\map \Pr \Omega = 1$.

• Results about the Poisson distribution can be found here.

Source of Name

This entry was named for Siméon-Denis Poisson.

Technical Note

The $\LaTeX$ code for \(\Poisson {\lambda}\) is \Poisson {\lambda} .

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

\Poisson \lambda

Sources

• 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $\S 2.2$: Examples: $(8)$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): distribution
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Poisson distribution
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): distribution
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Poisson distribution
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Poisson 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

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• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 39$: Probability Distributions: Poisson Distribution: $39.2$