Name: __________________________
MATH 321 Assignment (Due: Feb. 15th, 2016)
Directions: Students may work in groups of 1, 2, or 3, but only one submission per group. Show all work
and intermediate steps. This is especially important for answers found using a calculator. For each problem,
it will help to identify the appropriate random variable, i.e., distribution, to use. Please feel free to use:
• binomial cdf calculator (http://stattrek.com/online-calculator/binomial.aspx)
• Poisson cdf calculator (http://stattrek.com/online-calculator/poisson.aspx)
Project for ZagStats Inc.
Xavier Perry-Mentor
Data Enthusiasts International
975 Ronald Fisher Dr.
Spokane, WA 99201
ZagStats Inc.
1122 Cura Personalis Blvd.
Spokane, WA 99258
Dear Sirs and/or Madams,
First of all, thank you for helping us with our probability problem. A client of ours, Frequentist Flyers,
studies airport logistics. They are interested in determining how many tickets should be sold for a particular
flight. For simplicity, we’ll focus on an airplane that sits 150 passengers.
An airline wishes to sell as many tickets as possible, but, of course, it is not that simple. For instance, if the
airline sells 150 tickets, it is likely that some passengers will be unable to make the flight (due to weather,
health, etc.) and the airplane will be only partially full. On the other hand, if the airline sells more than 150
tickets, there is a chance that more passengers will arrive for the flight than the airplane can seat. When this
happens, the airline is responsible for compensating passengers that are bumped to other flights. Recent data
shows that, on average, 47 out of 50 people make their scheduled flights. The event that an individual misses
a particular flight is independent of other individuals, other flights, etc.
We request the following from you:
• Determine the maximum number of tickets, M , such that the probability of an overfull flight is less
than 0.05. What is this probability? hint: there is no nice formula; it’s sort of guess and check
• Use this number of tickets M in the following and assume that all M tickets are sold.
–
–
–
–
What is the expected number of passengers that will show up for a given flight?
What is the probability exactly 150 people will show for the flight?
What is the probability there will be at least one empty seat on the plane?
There are 20 flights on a particular day, each one with M tickets sold as described above. What is
the probability that more than one of these flights is overfull?
We look forward to your report and recommendations.
Sincerely,
X. Perry-Mentor
DEI
7
From Poisson to Exponential
Let Y have a Poisson distribution with rate ⁄ > 0, i.e.,
Y ≠⁄ y
]e ⁄
y = 0, 1, 2, ...
p(y; ⁄) =
y!
[
0
else.
Recall Y is the number of events that occur in a unit interval. Then, the distance X between consecutive
arrivals has an exponential distribution with rate ⁄, i.e.,
I
⁄e≠⁄x x Ø 0
f (x; ⁄) =
0
else.
Example: Pitting occurs in control rods used in nuclear reactors from continual surface degradation. The
number of pits in a 1 cm section of a control rod has a Poisson distribution with mean 4 pits. Consequently,
the distance between consecutive pits is exponentially distributed with mean 1/4 = 0.25 cm.
• Find the probability exactly 3 pits are found in a 1 cm section.
hint: Poisson RV
• A control rod is removed if it is found to have 6 pits (or more) in a randomly selected 1 cm section.
Find the probability at least 6 pits are found in this 1 cm section.
hint: Poisson RV
• A sample of 10 controls rods are measured for pitting (as described in the previous question).
Determine the probability that more than 4 of those rods are removed.
hint: binomial RV
• A pit was just discovered. Find the probability another pit will be found within the next 0.2 cm.
hint: exponential RV
• A pit was just discovered. Find the probability the next pit is more than 2 cm away.
hint: exponential RV
3