A television program reported that the U.S. (annual) birth rate is about 24 per
ID: 3227671 • Letter: A
Question
A television program reported that the U.S. (annual) birth rate is about 24 per 1000 people, and the death rate is about 8 per 1000 people. (a) Explain why the Poisson probability distribution would be a good choice for the random variable
r = number of births (or deaths) for a community of a given population size?
Frequency of births (or deaths) is a common occurrence. It is reasonable to assume the events are dependent. Frequency of births (or deaths) is a common occurrence. It is reasonable to assume the events are independent. Frequency of births (or deaths) is a rare occurrence. It is reasonable to assume the events are dependent. Frequency of births (or deaths) is a rare occurrence. It is reasonable to assume the events are independent.
(b) In a community of 1000 people, what is the (annual) probability of 7 births? What is the probability of 7 deaths? What is the probability of 14 births? 14 deaths? (Round your answers to four decimal places.)
(c) Repeat part (b) for a community of 1500 people. You will need to use a calculator to compute P(7 births) and P(14 births). (Round your answers to four decimal places.)
(d) Repeat part (b) for a community of 750 people. (Round your answers to four decimal places.)
Explanation / Answer
Solution:
(a)
Attributes of a Poisson Experiment:-
A Poisson experiment is a statistical experiment that has the following properties:
In a Poisson Distribution the probability of success gets smaller and smaller as the number of trials gets larger and larger.
In the above case, we have occurences of event is small . Therefore the probability of success is small.
So, in this case also the probability of success is smaller we can use Poisson distribution for better predictability of future events.
(b)
The average birth rate in US , 1 = 24/1000=0.024
The average death rate in US , 2 =8/1000=0.008
The probability distribution of a Poisson random variable X representing the number of successes occurring in a given time interval or a specified region of space is given by the formula:
P(X) =e-x / x!
The (annual) probability of 7 births , P(7 births) = e-0.024 * (0.024)7/7! = 8.884E-16 = 8.88 / 1016
The (annual) probability of 7 deaths , P(7 deaths) = e-0.008 * (0.008)7/7! = 4.1278E-19 =4.127/1019
The (annual) probability of 14 deaths , P(14 deaths) = e-0.008 * (0.008)14/14! =5.0E-41
(c)
The average birth rate on 1000 people is ,1 =0.024
The average birth rate on 1500 people is , 1 = (0.024/1000) * 1500 =0.036
The average death rate on 1000 people is ,2 =0.008
The average death rate on 1500 people is ,2 =(0.008/1000)*1500 =0.012
The (annual) probability of 7 births , P(7 births) = e-0.036 * (0.036)7/7! = 1.4998E-14
The (annual) probability of 7 deaths , P(7 deaths) = e-0.012 * (0.012)7/7! = 7.02E-18
The (annual) probability of 14 births , P(14 births) = e-0.036 * (0.036)14/14! =6.795E-32
The (annual) probability of 14 deaths , P(14 deaths) = e-0.012 * (0.012)14/14! =1.45E-38
(d)
The average birth rate on 1000 people is ,1 =0.024
The average birth rate on 750 people is , 1 = (0.024/1000) * 750=0.0018
The average death rate on 1000 people is ,2 =0.008
The average death rate on 750 people is ,2 =(0.008/1000)*750 =0.006
The (annual) probability of 7 births , P(7 births) = e-0.018 * (0.018)7/7! = 1.193E-16
The (annual) probability of 7 deaths , P(7 deaths) = e-0.006 * (0.006)7/7! = 5.52E-20
The (annual) probability of 14 births , P(14 births) = e-0.018 * (0.018)14/14! =4.22E-36
The (annual) probability of 14 deaths , P(14 deaths) = e-0.006 * (0.006)14/14! =8.93E-43
The (annual) probability of 14 births , P(14 births) = e-0.024 * (0.024)14/14! =2.35E-34Related Questions
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