Foundations of Data Science

Poisson distribution
The Poisson distribution describes the probability of k events happening in a unit of
time when the average rate per unit of time is λ. Divide the unit of time into n segments.
When n is large enough, each segment is sufficiently small so that the probability of two
events happening in the same segment is negligible. The Poisson distribution gives the
probability of k events happening in a unit of time and can be derived from the binomial
distribution by taking the limit as n → ∞.
Let p = λ n . Then
( ( n λ
Prob(k successes in a unit of time) = lim
n→∞ k)
n
= lim
n→∞
n (n − 1) · · · (n − k + 1)
k!
λ k
= lim
n→∞ k! e−λ
( λ
n) k (
1 − λ n
) k (
1 − λ ) n−k
n
) n (
1 − λ ) −k
n
In the limit as n goes to infinity the binomial distribution p (k) = ( n
k)
p k (1 − p) n−k becomes
the Poisson distribution p (k) = e . The mean and the variance of the Poisson
−λ λk
k!
distribution have value λ. If x and y are both Poisson random variables from distributions
with means λ 1 and λ 2 respectively, then x + y is Poisson with mean m 1 + m 2 . For large
n and small p the binomial distribution can be approximated with the Poisson distribution.
The binomial distribution with mean np and variance np(1 − p) can be approximated
by the normal distribution with mean np and variance np(1−p). The central limit theorem
tells us that there is such an approximation in the limit. The approximation is good if
both np and n(1 − p) are greater than 10 provided k is not extreme. Thus,
( ) ( ) k ( ) n−k n 1 1 ∼= 1
√ e − (n/2−k)2
1
2 n .
k 2 2 πn/2
This approximation is excellent provided k is Θ(n). The Poisson approximation
( n
k)p k (1 − p) k ∼ = e
−np (np)k
is off for central values and tail values even for p = 1/2. The approximation
( ) n
p k (1 − p) n−k ∼ 1 = √ e − (pn−k)2
pn
k
πpn
is good for p = 1/2 but is off for other values of p.
k!
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