Poisson Distribution

We have a Poisson process if:

• Independence: the number of occurrences in non-overlapping intervals are independent
• Individuality: the probability of 2 or more events occurring in a short interval is close to 0
• Homogeneity/uniformity: events occur at a uniform or homogenous rate over time so that the probability of one occurrence in an interval is

In a Poisson process with a rate of occurrence in a time interval length , given random variable :

where is the number of events in a fixed unit of time and .

Note that while we defined this in terms of time, we may have any rate (i.e frequency with area)

All whole numbers (natural numbers and 0)

Given :

(see Euler's Number, Factorial)

Given :

Poisson models are useful in modelling events where is large, and is small, and we are interested in the number of successes

1. no. texts in an hour
2. no. chocolate chips in a cookie
3. no. earthquakes per year

If and and , then:

where is like (the rate), and is like .

Cystic fibrosis is a genetic disorder that causes thick mucus to form in the lungs and other organs. Approximately 1 in every 2 500 Caucasian babies is born with cystic fibrosis In a random sample sample of 500 Caucasian babies, what is the probability that exactly 2 have cystic fibrosis?

solution
With binomial distribution:
:

With poisson distribution:

Suppose during software testing, a certain bug causes errors. The number of errors caused in a time period of length is assumed to have a Poisson distribution. You are given that in 2 hours, the bug caused one error on average.

Note that .

1. What is the probability that the bug will cause 3 errors in the next 2 hours?
   We have

2. What is the probability that the bug will cause at least 1 error in the next 2 hours?
   Again, . This time we find the complementary event.

3. How long does testing need to continue such that there is a 95% probability that at least one error has been caused?
   . We know from part (2) that , and :

(see log laws)
4. You are given another bug that produces no errors in the first 5 hours of testing with a probability of 50%. What is the probability that this bug will produce 2 errors in 10 hours?
We are given . To find rate of errors:

Now we find the probability of 2 errors in 10 hours:

Suppose 911 calls arrive at a rate of 3 per minute and follow a Poisson process.

1. Find the probability that exactly 2 calls arrive in 30 seconds
2. Find the probability of getting 6 calls in 2.5 minutes
3. Find the probability of 2 calls in the 1st minute given there were 6 calls in the first 2.5 minutes

solution
1.
Let be the number of calls in 30 seconds

Let be the number of calls in 2.5 minutes

We must find :