The Binomial distribution and Poisson distribution

Criteria for Poisson distribution

1. Events must occur independently 2. Events must occur singly 3. The average rate of occurrences must remain constant 4. There

Criteria for binomial distribution

1. The number of trials is fixed 2. Each trial has exactly two outcomes (success/failure) 3. The probability in each trial is fixed 4. Trials are independent of one another

Mean of the binomial distribution

E(x) = np

'Greater than' probability rule

P ( X ≥ x ) = 1 - P (X < x) = 1 - P(X≤ (x-1))

Poisson probability distribution

P (X = r) = e^(-λ) λ^(r) ÷ r!

Recurrence relation

P (X = r) = λ P(X = x-1) ÷ r

Binomial probability distribution

P(X=r) = ⁿCr pr q(n-r) can be used when values given are not in the tables

Binomial tables

Show P ( X ≤ x) The tables only show p values up to 0.5. TO work out p values higher than this you need to reverse the roles of success and failure.

When is the Poisson distribution used?

To find the probability of independent event which occur randomly, but at an overall constant rate.

When is the Binomial distribution used?

To model situations where the same 'trial' is carried out repeatedly a fixed number of times.

Variance of the binomial distribution

Var (x) = npq = σ² (How spread out the data is)

Notation

X∼B(n , p) n = no. of trials p = probability of 'success' in any one trial q = probability of 'failure' in any one trial (q = 1 - p)

Notation

X∼Po (λ) λ = the mean number of occurrences in the time period

Modelling with the binomial distribution

p = 0.5 - the distribution is symmetrical p > 0.5 - the distribution is negatively skewed p < 0.5 - the distribution is positively skewed