Homework sheet

The Poisson VariableIn a Poisson process, we count the total number of occurrances of a specified event in
an interval of time, distance, area, etc. Examples include:
• the number of typographical errors in a manuscript
• phone calls made to 911
• cracks on a paved highway
• births in a maternity ward
The variable X counts the total number of occurrances.
The Greek letter λ (lambda) represents the average number of occurrances per unit
of time, distance, area, etc. Examples are:
• the average number of typographical error per page of manuscript
• the average number of phone calls made to 911 per hour
• the average number of cracks per mile of highway
• the average number of births per day in a maternity ward
The probability of exactly r occurrences in an interval of time, distance, area, etc., where
λ is the mean number of occurrences per unit is:
e−λ · λr
P (X = r) =
r!
• The possible number of occurrances X = 0, 1, 2, …
• The number e ≈ 2.718
Example: There are an average of 200 typographical errors randomly distributed in a
500-page manuscript.
• Find the average number of errors per page. This is called the unit rate.
Using a proportion,
x errors
200 errors
=
500 pages
1 page
−→
x = 0.4 errors/page
• Find the probability that 1 single page contains exactly 2 errors.
Our number of units in this problem is 1 page, so for one page, λ = 0.4 errors
P (X = 2) =
e−0.4 · (0.4)2
= 0.054
2!
• Find the probability that a group of 5 pages contains a total of exactly 3 errors.
Our number of units in this problem is 5 pages, so for 5 pages,
λ = (0.4 errors/page) · (5 pages) = 2 errors
e−2 · (2)3
P (X = 3) =
= 0.180
3!
• Find the probability that a group of 3 pages has a total of less than 2 errors
Our number of units in this problem is 3 pages, so for 3 pages,
λ = (0.4 errors/page) · (3 pages) = 1.2 errors
P (X < 2) = P (X = 0) + P (X = 1) e−1.2 · (1.2)0 e−1.2 · (1.2)1 + = 0.301 + 0.361 = 0.662 = 0! 1! • Find the probability that a group of 6 pages has at least one error. Our number of units in this problem is 6 pages, so for 6 pages, λ = (0.4 errors/page) · (6 pages) = 2.4 errors P (X ≥ 1) = 1 − P (X = 0) = e−2.4 · (2.4)0 = 1 − 0.091 = 0.909 0! Complete the following: 1. Births in a hospital occur according to a Poisson process at an average rate of 43.2 births every 24 hours. • Find the unit rate (average number of births per hour) • Find the probability of exactly 1 birth occurring in the next hour. • Find the probability of exactly 3 births occurring in the next 2 hours • Find the probability of no births occurring in the next 30 minutes. 2. The number of cracks in a length of highway that require repair follows a Poisson distribution with an average of two cracks per five miles. • Find the unit rate (average number of cracks per mile) • Find the probability of no more than 2 cracks needing repair in a stretch of 8 miles. • Find the probability of at least one crack needing repair in a stretch of 3.5 miles. Discrete Variable/Normal Distribution Challenge Worksheet 1. A discrete random variable X has the probability distribution shown below, with one value missing. The standard deviation of the variable is σ = 3.816. Find the missing value of X, rounded to the nearest integer. Hint: You will need the quadratic formula! X P(X) 0 0.5 4 0.3 ??? 0.2 2. The NYPD reports that the average time it takes to respond to an emergency call in New York City is 25 minutes. Assume response time is normally distributed. Response time is considered ”too slow” if the time exceeds 30 minutes, and about 11% of all repsonse times are categorized as ”too slow”. What is the standard deviation of response time? 1