Probability Distributions

Question 1

Under what conditions is the Poisson distribution valid?

Answer 1

• The events must occur randomly
• The events must be independent of each other
• The average rate of the events must be constant

Question 2

What does the geometric distribution model?

Answer 2

The random variable: number of trials up to and including the first success

Question 3

What conditions must hold for the geometric distribution to be valid

Answer 3

• There are trials where all outcomes can be classified into two distinct categories
• The trials are independent
• The probability of success, p, is constant for each trial
• There is no upper limit to the number of trials

Question 4

For a geometric distribution, what does P(X>x) = ?

Answer 4

P(X>x) = (1-p)^x

Question 5

What are the key phrases in a question which indicate you should use a Poisson distribution?

Answer 5

• Average rate of events
• Events occuring at a constant rate
• NEVER a fixed number of trials

Question 6

What properties of your data indicate the Poisson might be a good model?

Answer 6

The mean and the variance are roughly equal (and therefore the standard deviation is the square root of this)

Question 7

If the number of butterflies seen in 10 minutes is x and you are using a poisson to model the number of butterflies in 1 hour, what is λ?

Answer 7

λ = 6x

Question 8

What can be said about the sum of two independent variables which both follow a Poisson distribution?

Answer 8

If two independent variables both follow a Poisson distribution, then so does their sum.