Poisson Distribution

$X\sim \text{Poisson}(\lambda t)$

• A Poisson variable represents the number of occurrences of an event in $t$ units of time that has an average occurrence of $\lambda$ occurrences per unit time
• Analogous to binomial distributions in the discrete world
• The $\mu$ below $=$ $\lambda t$
• $p_X(x)=e^{-\mu }\frac{{\mu }^x}{x!},x\in \{0,1,2,\dots \}$
• $F_X(x)=$
• $M_X(t)=e^{\mu (e^t-1)}$
• $\mathbb{E}[X]=\mu$
• $\text{var}(X)=\mu$

Example

• A radioactive particle emits a 10 photon every 5 seconds on average. What’s the probability that no particles are emitted over the timespan of one second?
  • $\lambda =\frac{10}{5}=2\to p_x(0)=e^{-2}\frac{2^0}{0!}=\frac{1}{e^2}\approx 0.135$

Applications

• Poisson Scattering

Derivation

• Can be derived from the binomial distribution when assuming $p\ll 1$ and $np$ remains finite (and even somewhat small) as $n\to \infty$