Math 160 - Cooley
TI Calculator Handout #4
OCC
Using the Poisson Distribution on the TI–84+
The Poisson Distribution is a discrete probability distribution that was developed by the French mathematician Simeon
Denis Poisson in 1837. It is used to calculate the frequency (probability) that a specified event occurs during a particular
period of time.
Each Poisson Distribution uniquely corresponds to a parameter called  (lambda), where:    and    .
The Poisson Distribution on the TI–84+ exists in two forms:
 Probability Density Function: poissonpdf(
 Cumulative Distribution Function: poissoncdf(
Both the PDF and CDF functions require the same initial information in order to calculate the probability. You will need
to specify the parameter  and the x–value.
Poisson PDF
Poisson CDF
Used to calculate EXACTLY x. Think P( X  x) .
Used to calculate AT MOST x. Think P( X  x) .
Function name: poissonpdf(
Function name: poissoncdf(
Note: The difference between the PDF function and CDF function is that the CDF is a cumulative sum that calculates all
probabilities less than or equal to the value of x.
Consider for a particular parameter  the difference between: poissonpdf(  , 3) and poissoncdf(  , 3 ).
 poissonpdf(  , 3 )  P( X  3) represents the probability that EXACTLY 3 specified events occur during a
particular period of time.
 poissoncdf(  , 3 )  P( X  3)  P( X  0)  P( X  1)  P( X  2)  P( X  3) represents the probability that
AT MOST 3 specified events occur during a particular period of time. Thus, for a fixed value of x:
The TI 84+ can only calculate P( X  x) and P( X  x) , yet, there are a few other situations that we encounter. Here is
the summary of syntax for those situations:
** SUMMARY OF SYNTAX **
SITUATION
1)
2)
3)
4)
P( X  x)
P( X  x )
P( X  x )
P(a  X  b)
SYNTAX
poissonpdf(  , x )
poissoncdf(  , x )
1 ‒ poissoncdf(  , x ‒ 1 )
poissoncdf(  , b ) ‒ poissoncdf(  , a ‒ 1 )
Example: Desert Samaritan Hospital keeps record of emergency room (ER) traffic. Those records indicate that the
number of patients arriving between 6:00 PM and 7:00 PM has a Poisson distribution with parameter   6.9 .
Determine the probability, that on a given day, the number of patients who arrive at the emergency room
between 6:00 PM and 7:00 PM will be
a) exactly 4
b) at most 2
c) 6 or greater
d) between 4 and 10 inclusive
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Math 160 - Cooley
Solution
1a
TI Calculator Handout #4
OCC
We want the probability that the number of patients is exactly 4. Thus, we want P( X  4) , which is
a simple pdf (probability density function) on the TI-84+.
TI-83+, TI-84+ (2.53MP and earlier)
TI-84+ (2.55MP)
▒ Key in: 2nd DISTR select poissonpdf( ENTER
▒ Key in: 2nd DISTR select poissonpdf( ENTER
This should be the screen you see.
This should be the screen you see.
The syntax for: poissonpdf( is
▒ Key in:
6.9 ENTER 4 ENTER
poissonpdf(  , x )
▒ Key in:
6.9
, 4 )
ENTER
Your cursor is on Paste, so, press ENTER again.
Thus, the probability, that on a given day, the number of
patients who arrive at the emergency room between
6:00 PM and 7:00 PM will be exactly 4 is
approximately 0.0952.
Press ENTER one more time.
Thus, the probability, that on a given day, the number of
patients who arrive at the emergency room between
6:00 PM and 7:00 PM will be exactly 4 is
approximately 0.0952.
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Math 160 - Cooley
Solution
1b
TI Calculator Handout #4
OCC
We want the probability that the number of patients is at most 2. Thus, we want P( X  2) , which is
a simple cdf (cumulative distribution function) on the TI-84+.
TI-83+, TI-84+ (2.53MP and earlier)
TI-84+ (2.55MP)
▒ Key in: 2nd DISTR select poissoncdf( ENTER
▒ Key in: 2nd DISTR select poissoncdf( ENTER
This should be the screen you see.
This should be the screen you see.
The syntax for: poissoncdf( is
▒ Key in:
6.9 ENTER 2 ENTER
poissoncdf(  , x )
▒ Key in:
6.9
, 2 )
ENTER
Your cursor is on Paste, so, press ENTER again.
Thus, the probability, that on a given day, the number of
patients who arrive at the emergency room between
6:00 PM and 7:00 PM will be at most 2 is
approximately 0.0320.
Press ENTER one more time.
Thus, the probability, that on a given day, the number of
patients who arrive at the emergency room between
6:00 PM and 7:00 PM will be at most 2 is
approximately 0.0320.
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Math 160 - Cooley
Solution
1c
TI Calculator Handout #4
OCC
We want the probability that the number of patients is 6 or greater. Thus, we want P( X  6) . Since
the calculator calculates up to at most a particular value and not greater or greater than or equal to,
then we need to rewrite our inequality of strictly ≤ signs, so that we can answer the question
correctly. So, P( X  6)  1  P( X  6) . Since we are dealing with a Poisson distribution, then we
know we are taking on discrete values. Thus, P( X  6)  1  P( X  6)  1  P( X  5) .
{That’s because P( X  6)  P( X  5) }. Now, we just need to calculate 1  P( X  5) on the TI-84+.
TI-83+, TI-84+ (2.53MP and earlier)
TI-84+ (2.55MP)
▒ Key in: 1 then ‒ &
▒ Key in: 1 then ‒ &
▒ Key in: 2nd DISTR select poissoncdf( ENTER
▒ Key in: 2nd DISTR select poissoncdf( ENTER
▒ Key in: 6.9
▒ Key in: 6.9 ENTER 5 ENTER&ENTER
, 5 ) ENTER
Thus, the probability, that on a given day, the number of
patients who arrive at the emergency room between
6:00 PM and 7:00 PM will be 6 or greater is
approximately 0.6863.
Solution
1d
Thus, the probability, that on a given day, the number of
patients who arrive at the emergency room between
6:00 PM and 7:00 PM will be 6 or greater is
approximately 0.6863.
We want the probability that the number of patients is between 4 and 10 inclusive. Thus, we want
P(4  X  10) , which is equivalent to the statement P( X  10)  P( X  4) which is equivalent to
P( X  10)  P( X  3) . So, we need to find the difference of two simple cdfs on the TI-84+.
TI-83+, TI-84+ (2.53MP and earlier)
TI-84+ (2.55MP)
▒ Key in: 2nd DISTR select poissoncdf( ENTER
▒ Key in: 2nd DISTR select poissoncdf( ENTER
▒ Key in: 6.9
, 10 )
▒ Key in: 6.9 ENTER 10 ENTER&ENTER
▒ Key in: ‒
(Note: minus sign, not negative sign)
▒ Key in: ‒
(Note: minus sign, not negative sign)
▒ Key in: 2nd DISTR select poissoncdf( ENTER
▒ Key in: 2nd DISTR select poissoncdf( ENTER
▒ Key in: 6.9
▒ Key in: 6.9 ENTER 3 ENTER&ENTER&ENTER
, 3 ) ENTER
Thus, the probability, that on a given day, the number of
patients who arrive at the emergency room between
6:00 PM and 7:00 PM will be between 4 and 10
inclusive is approximately 0.8213.
Thus, the probability, that on a given day, the number of
patients who arrive at the emergency room between
6:00 PM and 7:00 PM will be between 4 and 10
inclusive is approximately 0.8213.
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Math 160 - Cooley
TI Calculator Handout #4
OCC
Sample TI Calculator Quiz on Poisson Distribution
Example:
The number of calls received by a car towing service in an hour has a Poisson distribution with parameter  = 1.64. Find
each of the following using a TI-calculator. Round all answers to three decimal places.
Find the probability that in a randomly selected hour the number of calls is:
a) exactly 2
a) ________________________
b) at most 2
b) ________________________
c) 5 or greater
c) ________________________
d) between 2 and 4 inclusive.
d) ________________________
Solution:
a) .261
b) .773
c) .026
d) .462
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