Atlanta’s Hartsfield-Jackson International Airport is the busiest airport in the world. On average there are 2,500 arrivals and departures each day.
How many airplanes arrive and depart the airport per hour?
What is the probability that there are exactly 100 arrivals and departures in one hour?
What is the probability that there are at most 100 arrivals and departures in one hour?
Let
X = the number of airplanes arriving and departing from Hartsfield-Jackson in one hour. The average number of arrivals and departures per hour is
$\frac{2,500}{24}$ ≈ 104.1667.
X ~
P (104.1667), so
P (
x = 100) = poissonpdf(104.1667, 100) ≈ 0.0366.
P (
x ≤ 100) = poissoncdf(104.1667, 100) ≈ 0.3651.
The Poisson distribution can be used to approximate probabilities for a binomial distribution. This next example demonstrates the relationship between the Poisson and the binomial distributions. Let
n represent the number of binomial trials and let
p represent the probability of a success for each trial. If
n is large enough and
p is small enough then the Poisson approximates the binomial very well. In general,
n is considered “large enough” if it is greater than or equal to 20. The probability
p from the binomial distribution should be less than or equal to 0.05. When the Poisson is used to approximate the binomial, we use the binomial mean
μ =
np . The variance of
X is
σ^{2} =
μ and the standard deviation is
σ =
$\sqrt{\mu}$ . The Poisson approximation to a binomial distribution was commonly used in the days before technology made both values very easy to calculate.
On May 13, 2013, starting at 4:30 PM, the probability of low seismic activity for the next 48 hours in Alaska was reported as about 1.02%. Use this information for the next 200 days to find the probability that there will be low seismic activity in ten of the next 200 days. Use both the binomial and Poisson distributions to calculate the probabilities. Are they close?
Let
X = the number of days with low seismic activity.
Using the binomial distribution:
P (
x = 10) = binompdf(200, .0102, 10) ≈ 0.000039
Using the Poisson distribution:
Calculate
μ =
np = 200(0.0102) ≈ 2.04
P (
x = 10) = poissonpdf(2.04, 10) ≈ 0.000045
We expect the approximation to be good because
n is large (greater than 20) and
p is small (less than 0.05). The results are close—both probabilities reported are almost 0.
On May 13, 2013, starting at 4:30 PM, the probability of moderate seismic activity for the next 48 hours in the Kuril Islands off the coast of Japan was reported at about 1.43%. Use this information for the next 100 days to find the probability that there will be low seismic activity in five of the next 100 days. Use both the binomial and Poisson distributions to calculate the probabilities. Are they close?
Let
X = the number of days with moderate seismic activity.
Using the binomial distribution:
P (
x = 5) = binompdf(100, 0.0143, 5) ≈ 0.0115
Using the Poisson distribution:
Calculate
μ =
np = 100(0.0143) = 1.43
P (
x = 5) = poissonpdf(1.43, 5) = 0.0119
We expect the approximation to be good because
n is large (greater than 20) and
p is small (less than 0.05). The results are close—the difference between the values is 0.0004.
The lower class boundary is found by subtracting 0.5 units from the lowerclass limit and the upper class boundary is found by adding 0.5 units to the upper class limit. The difference between the upper and lowerboundaries of any class.
Ekene
The lower class boundary is found by subtracting 0.5 units from the lowerclass limit and the upper class boundary is found by adding 0.5 units to the upper class limit. The difference between the upper and lowerboundaries of any class.