Post on 24-Dec-2015
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