Poisson Distribtution

Question 1

Conditions for Poisson distribution

Answer 1

(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.

1. Each occurrence is independent of other occurrences
2. Events cannot occur simultaneously
3. Events occur at random and are unpredictable
4. For a small interval the probability of the event occurring is proportional to the size of the interval

Question 2

Poisson probability formula and notation

Answer 2

Given in formula sheet

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

Question 3

Required intervals

Answer 3

The value of 𝜆 must correspond to the required interval

Question 4

Variance of a Poisson distribution (4)

Answer 4

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

The mean and variance of poisson distribution are equal

Question 5

Poisson distribution as an approximation to the binomial distribution

Answer 5

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 normal distribution, and steps

Answer 6

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 lookup in table for answers
Solve