Poisson Distribution

The Poisson Distribution is used for• Discrete events (those you can count)• In a continuous but finite interval of time and space

The events can be counted and occur randomly at any time or place. The is no upper limit of events.

λ (lambda) is the measurement we will use in the

formula and is the mean number of occurrences

Examples:

X= the number of earthquakes in NZ over 6.0 on the Richter scale per year. λ = 4

X= the number of defects in a 5km telecommunications cable. λ = 3.65

Poisson DistributionPoisson Probability Formula

Example: The average number of scholarships gained each year is 6. Calculate the probability that there are exactly 4 scholarships gained in any one year.

x = 4

λ = 6

= 0.1339

Poisson Distribution

Calculate:

Answers

1. 0.224

2. 0.1733

3. 0.0919

Poisson Distribution

The number of meteorite strikes per year in Australia can be modelled by a Poisson random variable with a mean of 1.5. Is it more likely that there will be 1 or 2 strikes in a randomly chosen year.

λ = 1.5 , x = 1

= 0.3347

= 0.2510

More likely to be 1 strike

Poisson Distribution

Over a 10 year period there have been a total of 34 faults lasting more than 1 second in an electrical network. The number of faults can be modelled by a Poisson distribution.

Calculate the probability that there are 2 faults in one year.

λ = 34/10 = 3.4, x = 2

= 0.1929

Poisson Distribution

A car wash operator counts the number of cars arriving at the premises and finds that, on average, there are 7 per hour. Assuming that this number has a Poisson distribution, find the probability that there are no cars in the car wash in any given quarter hour.

λ = 7/4 = 1.75

x = 0

= 0.1738

Poisson Distribution

Using the tables: X is a random variable with mean 2. Find

a. P(X ≥ 3)

P = 0.3232

Poisson Distribution

Using the tables: X is a random variable with mean 2. Find

a. P(X ≤ 1)

P = 0.406

Poisson Distribution

The number of mature toheroa per square metre at a West Coast beach has a Poisson distribution with a mean of 1.6. When an area of 3 m2 is searched, what is the probability that fewer than 6 will be found?

λ = 3 x 1.6 = 4.8

x < 6 ie 0, 1, 2, …, 5

0.0082

0.0395

0.0948

0.1517

0.1820

0.1747

0.6509