Let Z be some random variable. Then associated with Z is a probability distribution function that assigns probabilities to the different outcomes Z can take.

If Z is discrete, then its distribution is called a probability mass function, which measures the probability Z takes on the value k: P(Z=k). The probability mass function completely describes Z.

Poisson distribution

Z is Poisson-distributed if:

1P(Z=k) = \frac{\lambda^k e^{-\lambda}}{k!}, k=1,2,3,...

$\lambda$ is a parameter of the distribution which controls its shape. Larger values assign more probability to larger k, smaller values assign more probability to smaller k.

A Z with distribution Poisson is denoted as $Z ~ Poi(\lambda)$

Its expected value is equal to its parameter: $E[Z | \lambda] = \lambda$.

References

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