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Sagot :
Answer:
a) 0.39984 = 39.984% probability of no orders in five minutes.
b) 0.06563 = 6.563% probability of 3 or more orders in five minutes.
c) The length of time is 0.63 hours
Step-by-step explanation:
In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
In which
x is the number of sucesses
e = 2.71828 is the Euler number
[tex]\mu[/tex] is the mean in the given interval.
Orders arrive at a Web site according to a Poisson process with a mean of 11 per hour.
This means that [tex]\mu = 11h[/tex], in which h is the number of hours.
a) Probability of no orders in five minutes.
Five minutes means that [tex]h = \frac{5}{60} = \frac{1}{12}[/tex], so [tex]\mu = \frac{11}{12} = 0.9167[/tex]
This probability is P(X = 0). So
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
[tex]P(X = 0) = \frac{e^{-0.9167}*(0.9167)^{0}}{(0)!} = 0.39984[/tex]
0.39984 = 39.984% probability of no orders in five minutes.
b) Probability of 3 or more orders in five minutes.
This is:
[tex]P(X \geq 3) = 1 - P(X < 3)[/tex]
In which
[tex]P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)[/tex]
So
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
[tex]P(X = 0) = \frac{e^{-0.9167}*(0.9167)^{0}}{(0)!} = 0.39984[/tex]
[tex]P(X = 1) = \frac{e^{-0.9167}*(0.9167)^{1}}{(1)!} = 0.36653[/tex]
[tex]P(X = 2) = \frac{e^{-0.9167}*(0.9167)^{2}}{(2)!} = 0.168[/tex]
[tex]P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) = 0.39984 + 0.36653 + 0.168 = 0.93437[/tex]
[tex]P(X \geq 3) = 1 - P(X < 3) = 1 - 0.93437 = 0.06563[/tex]
0.06563 = 6.563% probability of 3 or more orders in five minutes.
c) Length of a time interval such that the probability of no orders in an interval of this length is 0.001.
This is h for which:
[tex]P(X = 0) = 0.001[/tex]
We have that:
[tex]P(X = 0) = e^{-\mu}[/tex]
And
[tex]\mu = 11h[/tex]
So
[tex]P(X = 0) = 0.001[/tex]
[tex]e^{-11h} = 0.001[/tex]
[tex]\ln{e^{-11h}} = \ln{0.001}[/tex]
[tex]-11h = \ln{0.001}[/tex]
[tex]h =-\frac{\ln{0.001}}{11}[/tex]
[tex]h = 0.63[/tex]
The length of time is 0.63 hours
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