Definition:Poisson Distribution

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Definition

The Poisson distribution is the discrete probability distribution of the Poisson process.

Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$ which is generated by a Poisson process.

Then $X$ has the Poisson distribution with parameter $\lambda$.

It is written:

$X \sim \Poisson \lambda$



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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:

$\Img X = \set {0, 1, 2, \ldots} = \N$
$\map \Pr {X = k} = \dfrac 1 {k!} \lambda^k e^{-\lambda}$


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

  • 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


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