Radioactive decay is Poisson
Consider a macroscopic sample of Uranium.
Each atom decays independently of the others, and the likelihood of a single atom popping off is very low; but the product of this likelihood by the total number of atoms is a moderate number.
So there is some constant average rate of atoms in the sample popping off, and the number of pops per minute follows a Poisson distribution.
Calls to a call center is Poisson
Consider a call center that receives help requests from users of a popular phone manufacturer.
The total number of users is very large, and the likelihood of any given user calling in a given minute is very small, but the product of these rates is moderate.
So there is some constant average rate of calls to the center, and the number of calls per minute follows a Poisson distribution.
Typos per page
A draft of a textbook has an average of 6 typos per page.
What is the probability that a randomly chosen page has
Answer: 0.849
Hint: study the complementary event.
Poisson calculation
Suppose
Solution
(1) Conditioning definition:
(2) Expand numerator:
(3) Simplify:
(4) Compute for denominator:
Arrivals at a post office
Client arrivals at a post office are modelled well using a Poisson variable.
Each potential client has a very low and independent chance of coming to the post office, but there are many thousands of potential clients, so the arrivals at the office actually come in moderate number.
Suppose the average rate is 5 clients per hour.
(a) Find the probability that nobody comes in the first 10 minutes of opening. (The cashier is considering being late by 10 minutes to run an errand on the way to work.)
(b) Find the probability that 5 clients come in the first hour. (I.e. the average is achieved.)
(c) Find the probability that 9 clients come in the first two hours.
Solution
(a)
(1) Convert rate for desired window.
Expect
Let
Seek
(2) Compute.
Formula:
Insert data and compute:
(b)
Rate is already correct.
Let
Compute the answer:
(c)
Convert rate for desired window.
Expect 10 clients every 2 hours.
Let
Compute the answer:
Notice that 0.125 is smaller than 0.175.