Math 160 - TI Calculator Handout #5
Math 160 - CooleyTI Calculator Handout #5OCC
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: .
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 CDFUsed to calculate EXACTLYx. Think . / Used to calculate AT MOSTx. Think .
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) represents the probability that EXACTLY 3 specified events occur during a particular period of time.
poissoncdf(, 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 and , yet, there are a few other situations that we encounter. Here is the summary of syntax for those situations:
** SUMMARY OF SYNTAX **SITUATIONSYNTAX
1) poissonpdf(, x )
2) poissoncdf(, x )
3) 1 ‒ poissoncdf(, x ‒ 1 )
4) 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 .
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 4b) at most 2 c)6 or greaterd)between 4 and 10 inclusive
Solution1a / We want the probability that the number of patients is exactly 4. Thus, we want , 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: 2ndDISTR select poissonpdf( ENTER
/ ▒ Key in: 2ndDISTR select poissonpdf( ENTER
This should be the screen you see.
/ This should be the screen you see.
The syntax for: poissonpdf( is
poissonpdf(, x)
▒ Key in: 6.9 , 4 ) 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 exactly 4 is approximately 0.0952. / ▒ Key in: 6.9 ENTER 4 ENTER
Your cursor is on Paste, so, press ENTER again.
PressENTER 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.
Solution
1b / We want the probability that the number of patients is at most 2. Thus, we want , 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: 2ndDISTR select poissoncdf( ENTER
/ ▒ Key in: 2ndDISTR select poissoncdf( ENTER
This should be the screen you see.
/ This should be the screen you see.
The syntax for: poissoncdf( is
poissoncdf(, x)
▒ Key in: 6.9 , 2 ) 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 at most 2 is approximately 0.0320. / ▒ Key in: 6.9 ENTER 2 ENTER
Your cursor is on Paste, so, press ENTER again.
PressENTER 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.
Solution
1c / We want the probability that the number of patients is 6 or greater. Thus, we want. 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, . Since we are dealing with a Poisson distribution, then we know we are taking on discrete values. Thus, .
{That’s because }. Now, we just need to calculate on the TI-84+.
TI-83+, TI-84+ (2.53MP and earlier) / TI-84+ (2.55MP)
▒ Key in: 1 then ‒
▒ Key in: 2ndDISTR select poissoncdf( ENTER
▒ Key in: 6.9 , 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. / ▒ Key in: 1 then ‒
▒ Key in: 2ndDISTR select poissoncdf( ENTER
▒ Key in: 6.9 ENTER 5 ENTERENTER
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 / We want the probability that the number of patients is between 4 and 10 inclusive. Thus, wewant , which is equivalent to the statement which is equivalent to . 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: 2ndDISTR select poissoncdf( ENTER
▒ Key in: 6.9 , 10 )
▒ Key in: ‒ (Note: minus sign, not negative sign)
▒ Key in: 2ndDISTR select poissoncdf( ENTER
▒ Key in: 6.9 , 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. / ▒ Key in: 2ndDISTR select poissoncdf( ENTER
▒ Key in: 6.9 ENTER 10 ENTERENTER
▒ Key in: ‒ (Note: minus sign, not negative sign)
▒ Key in: 2ndDISTR select poissoncdf( ENTER
▒ Key in: 6.9 ENTER 3 ENTERENTERENTER
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.
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 2a) ______
b)at most 2b) ______
c)5 or greaterc) ______
d)between 2 and 4 inclusive.d) ______
Solution:
a) .261b) .773c) .026d) .462
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