Probability - Poisson Process Please show details so I may learn the process. Th

Question

Probability - Poisson Process

Please show details so I may learn the process. Thank you so much.

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People are leaving a line at a constant rate of 2 per minute.

a) What is the probability that at least 2 people leave during a 4 minute timespan?

b) How many people do you expect to leave, on average, during a 10 minute timespan?

c) What is the probability that the number of people that leave during the 10 minute timespan is within 1 from the expected number?

d) What is the expected waiting time between the second person that leaves and the third one?

e) What is the probability that you have to wait longer than one minute for the first person to leave?

f) If you watch for a while and the people keep leaving and no one enters the line, what might be the reason this process may stop being a Poisson process?

Explanation / Answer

a)

There are r = 2/minute, so for 4 minutes,

u = r t = (2/min)(4 min) = 8 people

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative poisson distribution table or technology, matching          
          
u = the mean number of successes =    8      
          
x = our critical value of successes =    2      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   1   ) =    0.003019164
          
Thus, the probability of at least   2   successes is  
          
P(at least   2   ) =    0.996980836 [ANSWER]

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b)

There are r = 2/minute, so for 10 minutes,

E(x) = r t = (2/min)(10 min) = 20 people [ANSWER]

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c)

That means from 19 to 21 inclusive.

Note that P(between x1 and x2) = P(at most x2) - P(at most x1 - 1)          
          
Here,          
          
x1 =    19      
x2 =    21      
          
Using a cumulative poisson distribution table or technology, matching          
          
u = the mean number of successes =    20      
          
          
Then          
          
P(at most    18   ) =    0.381421949
P(at most    21   ) =    0.643697648
          
Thus,          
          
P(between x1 and x2) =    0.262275699   [ANSWER]

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d)

The rate is 2 people/min, so the interval between 2 consecutive persons is on the average

u = 1 min/2 = 0.5 min or 30 seconds [ANSWER]  

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