Poisson distribution

Dependencies:

1. Random variable
2. Series expansion for e^x (incomplete)

Let $X$ be a random variable whose support is $\mathbb{Z}_{\ge 0}$. Then $X$ is said to be Poisson-distributed with parameter $\lambda$ (where $\lambda > 0$) iff \[ \Pr(X = k) = \frac{e^{-\lambda}\lambda^k}{k!}. \] We use $X \sim \operatorname{Poisson}(\lambda)$ to denote that $X$ is Poisson-distributed with parameter $\lambda$.

Proof that $X$ is a probability distribution

$\Pr(X = k) \ge 0$ for all $k$. \[ \sum_{k=0}^{\infty} \Pr(X = k) = \sum_{k=0}^{\infty} \frac{e^{-\lambda}\lambda^k}{k!} = e^{-\lambda} \sum_{k=0}^{\infty} \frac{\lambda^k}{k!} = e^{-\lambda} e^{\lambda} = 1. \]

Dependency for:

1. Poisson process

Info:

• Depth: 6
• Number of transitive dependencies: 13

Transitive dependencies:

1. /analysis/topological-space
2. /sets-and-relations/countable-set
3. /sets-and-relations/de-morgan-laws
4. σ-algebra
5. Generated σ-algebra
6. Borel algebra
7. Measurable function
8. Generators of the real Borel algebra (incomplete)
9. Measure
10. σ-algebra is closed under countable intersections
11. Probability
12. Random variable
13. Series expansion for e^x (incomplete)