Question 1088788: Sports Score Hot Line Calls (Optional) Sports Scores Hot Line receives, on the average, 8 calls per hour requesting the latest sports scores. The distribution is Poisson in nature. For any randomly selected hour, find the probability that the company will receive a. At least 8 calls b. 3 or more calls c. At most 7 calls. Full solution please
Answer by mathmate(429)   (Show Source):
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Sports Score Hot Line Calls (Optional) Sports Scores Hot Line receives, on the average, 8 calls per hour requesting the latest sports scores. The distribution is Poisson in nature. For any randomly selected hour, find the probability that the company will receive a. At least 8 calls b. 3 or more calls c. At most 7 calls. Full solution please
Solution:
Given it is a Poisson process, describing number of occurrences within a given period of time. The situation fits the requirements of the Poisson distribution.
The PMF of the distribution is given by:
P(k)=lambda^k*e^(-lambda)/k! ..........(1)
where lambda is the mean number of occurrences within a given time period
The CDF exists in closed form that involves gamma functions. We could get by using only PMF.
Here we have lambda=8 calls per hour (period is 1 hour)
Using equation (1) above, we calculate P(k) for k=0 to 8, using lambda=8:
Example: k=0, P(0)=8^0*e^-8/0!=0.000335
k P(k)
0 0.000335
1 0.002683
2 0.010735
3 0.028626
4 0.057252
5 0.091604
6 0.122138
7 0.139587
8 0.139587
Total=0.592547
With the above table, we can then answer the given questions.
(a) At least 8 calls
P(K>=8)=1-P(K<8)
=1-(P(0)+P(1)+P(2)...+P(7))
=1-(0.000335+0.002683+0.010735+0.028626+0.057252+0.091604+0.122138+0.139587)
=1-0.452961
=0.547039
(b) Three or more calls
P(K>=3)=1-P(K<3)
=1-(P(0)+P(1)+P(2))
=1-((0.000335+0.002683+0.010735)
=0.986246
(c) At most 7 calls
P(K<=7)=1-P(K>=8)
=1-0.547039
=0.452961
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