Poisson Distribution

Properties

useful model when events can be thought of as occurring randomly and independently over time/area/volume

number of chocolate chips in a cookie
number of radioactive decays of a substance in one second
poisson approximates the Binomial Distribution when $n \to \infty$ and $p \to 0$
reasonable if $n > 50$ and $n p < 5$
no natural upper bound, % goes down but could be possible to have a very large value of occurrences

Assumptions

Independence

number of occurrences in non-overlapping intervals needs to be independent

P [x (t_{1}, t_{2}) = x_{1}, x (t_{2}, t_{3} = x_{2})] = P (x (t_{1}, t_{2}) = x_{1}) \cdot P (x (t_{2}, t_{3}) = x_{2})

Individuality

the probability of more than one success during a very small time interval/region is negligible
events can only happen one at a time
for sufficiently short time periods of length $Δ t$ , the probability of 2 or more events occurring in the interval is close to zero i.e. events occur singly not in clusters.
- as $Δ t \to 0$ , the probability of two or more events in the interval of length $Δ t$ must go to zero faster than $Δ t \to 0$

Uniformity

events occur at a uniform or homogenous rate $λ$ over time so that the probability of one occurrence in an interval $(\vec{t}, t + Δ t)$ is approximately $λ Δ t$ for any value of t

Probability of a single success occurring in a very short time interval/region is given by $λ \cdot Δ t$

Probability Distribution Function

A random variable $X$ has a Poisson distribution $X \sim P o i s s o n (λ)$ if and only if it has the probability mass function

P(X=x)=f(x;\lambda)=\frac{\lambda {#xe} ^{-\lambda}}{x!}, x = 0,1,2\dots,\quad \quad\lambda>0

$f (x; λ) > 0$
$\sum_{x} f (x; λ) = 1$

E (X) = V a r (X) = μ = σ^{2} = λ

Moment Generating Function

M_{X} (t) = e^{λ (e^{t} - 1)} = e^{e^{t} λ} e^{- λ}

Relationship with Binomial

The poisson distribution is derived from the Binomial Distribution

If we consider intervals of time as being the independent events from the binomial, we can use its pdf and calculate a new function for when there are infinite intervals, or an infinitesimal small interval

The probability of one or no occurrences happening in the interval fsd

$X \sim B i n (n, p), n \to \infty, p \to 0, n p = λ, p = \frac{λ}{n}$

\begin{aligned} \lim_{n \to \infty} f (x; n, p) & = (\binom{n}{x}) p^{x} (1 - p)^{n - x} \\ = (\binom{n}{x}) {(\frac{λ}{n})}^{x} {(1 - \frac{λ}{n})}^{n - x} \\ = \frac{n (n - 1) (n - 2) \dots (n - x + 1) (n - x)!}{x! (n - x)!} \frac{λ^{x}}{n^{x}} {(1 - \frac{λ}{n})}^{n} {(1 - \frac{λ}{n})}^{- x} \\ = \frac{n (n - 1) (n - 2) \dots (n - x + 1)}{x!} \frac{λ^{x}}{n^{x}} {(1 - \frac{λ}{n})}^{n} {(1 - \frac{λ}{n})}^{- x} \\ = \frac{λ^{x}}{x!} {(1 - \frac{λ}{n})}^{n} {(1 - \frac{λ}{n})}^{- x} \frac{n (n - 1) \dots (n - x + 1)}{n^{x}} \\ = \frac{λ^{x}}{x!} lim_{n \to \infty} {(1 - \frac{λ}{n})}^{n} \cdot 1 \cdot 1 \cdot (1 - \frac{1}{n}) (1 - \frac{2}{n}) \dots (1 - \frac{x - 1}{n}) (x terms on num and denom) \\ = \frac{λ^{x}}{x!} e^{- λ} \cdot 1 \\ = \frac{λ^{x} e^{- λ}}{x!} = P \sim (x, λ) \end{aligned}

Examples

if:
- events are occurring independently in time (knowing when one event occurs gives no information about when another event will occur)
  - 1st = poisson, random
  - 2nd = clumped, not poisson
  - 3rd = grid/ordered/evenly distributed, not poisson
- the probability that an event occurs in a given length of time does not change through time (theoretical rate of events stays constant through time)
then X = number of events in a fixed unit of time and has a
- poisson distribution with probability mass function
  - with only one parameter
- for x = 0,1,2,...

Properties

Assumptions

Probability Distribution Function

Relationship with Binomial

Examples

for x = 0,1,2,...