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Text message users receive or send an average of 41.5 text messages per day.

  1. How many text messages does a text message user receive or send per hour?
  2. What is the probability that a text message user receives or sends two messages per hour?
  3. What is the probability that a text message user receives or sends more than two messages per hour?
  1. Let X = the number of texts that a user sends or receives in one hour. The average number of texts received per hour is 41.5 24 ≈ 1.7292.
  2. X ~ P (1.7292), so P ( x = 2) = poissonpdf(1.7292, 2) ≈ 0.2653
  3. P ( x >2) = 1 – P ( x ≤ 2) = 1 – poissoncdf(1.7292, 2) ≈ 1 – 0.7495 = 0.2505
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Atlanta’s Hartsfield-Jackson International Airport is the busiest airport in the world. On average there are 2,500 arrivals and departures each day.

  1. How many airplanes arrive and depart the airport per hour?
  2. What is the probability that there are exactly 100 arrivals and departures in one hour?
  3. What is the probability that there are at most 100 arrivals and departures in one hour?
  1. 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 2 , 500 24 ≈ 104.1667.
  2. X ~ P (104.1667), so P ( x = 100) = poissonpdf(104.1667, 100) ≈ 0.0366.
  3. 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 = μ . The Poisson approximation to a binomial distribution was commonly used in the days before technology made both values very easy to calculate.

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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.

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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.

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Questions & Answers

where do I start
Jeremy Reply
in a large restaurant an average of every 7 customers ask for water with the their meal. A random sample of 12 customers is selected, find the probability that exactly 6 ask for water with their meal
Rufai Reply
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Rufai
conditional probability
Ramesh
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Rufai
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Ramesh
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Rufai
this follows binomial distribution. p(X=6)=12C6*(0.6)^6*0.4^6 use this formula n find.
syeda
can you explain the cosidered variable in the formula
Divya
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syeda
n is number of customers
syeda
ncx*p^X*q^X?
Divya
q^n-x
syeda
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Divya
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Hoshyar
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Hoshyar
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khalid
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Shadow Reply
Descriptive statistics are brief descriptive coefficients that summarize a given data set, which can be either a representation of the entire or a sample of a population. Descriptive statistics are broken down into measures of central tendency and measures of variability (spread). 
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james Reply
the upper quartile of the population 10,12,14,16,18,20,25,15,11,11,17,is................?
Gach
The probability range is 0 to 1... but why we take it 0 to 1....
Muhammad Reply
because in probability 1 means success and 0 means failure and it cnnt be more or less than 1 and 0.
syeda
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khalid
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Hoshyar
The probability of an event will not be less than 0.  This is because 0 is impossible (sure that something will not happen).The probability of an event will not be more than 1. This is because 1 is certain that something will happen
Divya
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Megrina Reply
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you're asked to find page 40 and page 88 on that particular book.
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IMAGESNEWSVIDEOS A Dictionary of Computing. measures of location Quantities that represent the average or typical value of a random variable (compare measures of variation). They are either properties of a probability distribution or computed statistics of a sample. Three important measures are the mean, median, and mode.
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Kaynaat Reply
IMAGESNEWSVIDEOS A Dictionary of Computing. measures of location Quantities that represent the average or typical value of a random variable (compare measures of variation). They are either properties of a probability distribution or computed statistics of a sample. Three important measures are th
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Muhammad
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BITRUS Reply
given that the probability of
BITRUS
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There are two coins on a table. When both are flipped, one coin land on heads eith probability 0.5 while the other lands on head with probability 0.6. A coin is randomly selected from the table and flipped. (a) what is probability it lands on heads? (b) given that it lands on tail, what is the Condi
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0.5*0.5+0.5*0.6
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Enhance
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Enhance
the science of summarization and description of numerical facts
Enhance
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Source:  OpenStax, Introductory statistics. OpenStax CNX. May 06, 2016 Download for free at http://legacy.cnx.org/content/col11562/1.18
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