Poisson Distribution

Question 1

Introduction

Answer 1

The Poisson random variable 𝑋 is the total number of occurrences of the discrete events in some given continuous interval.

(a) Discrete event occurs, could occur at any time, and in theory is no upper limit on the number of occurrences.
(b) The interval is some continuous measurement, such as time, length or area.

𝑋 is the number of events of a phenomenon occurring randomly in time or space.
The single parameter 𝜆 is the mean number of occurrences of the event over the given interval.
𝜆 is a rate and 𝑋 represents the number of occurrences of the event.

Question 2

Poisson probability formula

Answer 2

P(X=x) = [ 𝑒^⁻𝜆 . 𝜆ˣ ] / x!
x = 0, 1, 2, ….

(*∑ x = # ([ 𝑒^⁻𝜆 . 𝜆ˣ ] / x!))
* upper limit
# lower limit

Notation
X~Po(𝜆) , X has a poisson distribution with mean 𝜆

𝜆 equals the mean number of events in the given interval

Poisson distribution is a proper probability distribution: the sum of all possibilities is 1.

The 𝜆 value must correspond to the required interval.

Question 3

Conditions for Poisson distribution

Answer 3

a) Each occurrence is independent of other occurrences.
(b) Events cannot occur simultaneously.
(c) Events occur at random and are unpredictable.
(d) For a small interval the probability of the event occurring is proportional to the size of the interval

If mean and variance are not approximately equal then the poisson distribution is not a suitable model

Question 4

Variance of Poisson distribution

Answer 4

X~Po(𝜆)
Mean = E(X) = µ= 𝜆
Variance = Var(x) = 𝜎² = 𝜆
Standard deviation = Sd(x) = 𝜎 = √𝜎

The mean and variance of poisson distribution are equal

Question 5

Poisson distribution as an approximation to the binomial distribution

Answer 5

For large values of n and very small values of p (almost zero), a binomial distribution with parameters n and p is closely approximated by a poisson distribution.

Binomial variance = npq ≈ np (because q is almost 1)
Binomial mean = np

If n > 50 and np < 5, then X can be approximated by poisson distribution
X ~ B(n, p) to X~Po(𝜆 = np)

Question 6

Poisson distribution as an approximation to the binomial distribution

Answer 6

For large values of 𝜆, Poisson probabilities approximately fit a normal distribution. A good guide for when a normal approximation to the Poisson is appropriate is when 𝜆 > 15.

If 𝑋~𝑃𝑜(𝜆) and if 𝜆 > 15 then 𝑋 can reasonably be approximated by the normal distribution, approximate 𝜇 = 𝜆 and 𝜎² = 𝜆.
A continuity correction must be applied.
∴ 𝑋~𝑃𝑜(𝜆) ⇒ 𝑋~𝑁(𝜇 = 𝜆, 𝜎² = 𝜆)

Steps:
Draw bell curve for visual understanding
Transcribe X~Po() and X~N() with continuity correction
Transcribe probability P(X > )
Solve Z
Solve for Z and look up in table for answers
Solve