Poisson Distribution & Problems

POISSON DISTRIBUTION
Definition:
The Poisson distribution is a discrete probability distribution of a random variable 𝑿 that satisfies the
following conditions:
➢ The experiment consists of counting the number of times 𝑿 an event occurs in a given interval.
The interval can be an interval of time, area, or volume.
➢ The probability of the event occurring is the same for each interval.
➢ The number of occurrences in one interval is independent of the number of occurrences in
other intervals.
Probability Mass Function

𝜆𝑥 𝑒 −𝜆
𝑃(𝑋 = 𝑥) =
𝑥!
where
𝑥 - is the number of times an event occurs in an interval
𝜆 - is the average number of events/successes per unit interval
𝑒 – Euler’s number/constant
Mean/Expectation
𝜇 = 𝐸[𝑋] = 𝜆
Variance
𝜎 2 = 𝑉𝑎𝑟[𝑋] = 𝜆
Standard Deviation
𝜎 = √𝜆
PROBLEMS
1.) The mean value for an event X to occur is 2 in a day. Find the probability of event X to occur
thrice in a day.
2.) A shop sells five pieces of shirt every day, then what is the probability of selling three shirts
today?
3.) Number of calls coming to the customer care center of a mobile company per minute is a
Poisson random variable with mean 5. Find the probability that no call comes in a certain
minute.
4.) Suppose a fast food restaurant can expect two customers every 3 minutes, on average. What is
the probability that four or fewer patrons will enter the restaurant in a 9 minute period?
5.) The average number of homes sold by the Acme Realty Company is 2 homes per day. What is
the probability that exactly 3 homes will be sold tomorrow?
6.) There are five students in a class and the number of students who will participate in annual day
every year is a Poisson random variable with mean 3. What will be the probability of more than
3 students participating in annual day this year?
7.) The average number of loaves of bread put on a shelf in a bakery in a half-hour period is 12.
What is the probability that the number of loaves, selected randomly, put on the shelf in five
minutes is three?
8.) An emergency room at a particular hospital gets an average of five patients per hour. A doctor
wants to know the probability that the ER gets more than five patients per hour. What’s your
response?
9.) Leah's answering machine receives about six telephone calls between 8 a.m. and 10 a.m. What
is the probability that Leah receives more than one call in the next 15 minutes?
10.) One nanogram of Plutonium-239 will have an average of 2.3 radioactive decays per second, and
the number of decays will follow a Poisson distribution. What is the probability that in a 2
second period there are exactly 3 radioactive decays?
11.) If electricity power failures occur according to a Poisson distribution with an average
of 3 failures every twenty weeks, calculate the probability that there will not be more than one
failure during a particular week.
12.) Vehicles pass through a junction on a busy road at an average rate of 300 per hour.
(a) Find the probability that none passes in a given minute.
(b) What is the expected number passing in two minutes?
(c) Find the probability that this expected number actually pass through in a given two-minute
period.
13.) Consider a computer system with Poisson job-arrival stream at an average of 2 per minute.
Determine the probability that in any one-minute interval there will be
(a) 0 jobs
(b) exactly 2 jobs
(c) at most 3 arrivals
14.) Births in a hospital occur randomly at an average rate of 1.8 births per hour.
(a) What is the probability of observing 4 births in a given hour at the hospital?
(b) What about the probability of observing more than or equal to 2 births in a given hour at
the hospital?
15.) According to Baydin, an email management company, an email user gets, on average, 147
emails per day.
(a) What is the probability that an email user receives exactly 160 emails per day?
(b) What is the probability that an email user receives at most 160 emails per day?
(c) What is the standard deviation?
16.) According to a recent poll by the Pew Internet Project, girls between the ages of 14 and 17 send
an average of 187 text messages each day.
(a) What is the probability that a teen girl sends exactly 175 texts per day?
(b) What is the probability that a teen girl sends at most 150 texts per day?
(c) What is the standard deviation?
17.) Text message users receive or send an average of 41.5 text messages per day.
(a) How many text messages does a text message user receive or send per hour?
(b) What is the probability that a text message user receives or sends two messages per hour?
(c) What is the probability that a text message user receives or sends more than two messages
per hour?
18.) Atlanta’s Hartsfield-Jackson International Airport is the busiest airport in the world. On average
there are 2,500 arrivals and departures each day.
(a) How many airplanes arrive and depart the airport per hour?
(b) What is the probability that there are exactly 100 arrivals and departures in one hour?
(c) What is the probability that there are at most 100 arrivals and departures in one hour?

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POISSON DISTRIBUTION

Probability Mass Function

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