The number N of calls arriving at a switchboard during a per

The number N of calls arriving at a switchboard during a period of one hour has the PMF What is the probability that at least 2 calls arrive within one hour? What is the probability that at most 3 calls arrive within one hour? What is the probability that the number of calls that arrive within one hour is greater than 3 but less than or equal to 6?

Solution

Here, we see a Poisson distribution of mean = 10.

a)

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 =    10      
          
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   ) =    0.000499399
          
Thus, the probability of at least   2   successes is  
          
P(at least   2   ) =    0.999500601 [answer]

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b)

Using a cumulative poisson distribution table or technology, matching          
          
u = the mean number of successes =    10      
          
x = the maximum number of successes =    3      
          
Then the cumulative probability is          
          
P(at most   3   ) =    0.010336051 [answer]

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c)

Note that P(between x1 and x2) = P(at most x2) - P(at most x1 - 1)          
          
Here,          
          
x1 =    4      
x2 =    6      
          
Using a cumulative poisson distribution table or technology, matching          
          
u = the mean number of successes =    10      
          
          
Then          
          
P(at most    3   ) =    0.010336051
P(at most    6   ) =    0.130141421
          
Thus,          
          
P(between x1 and x2) =    0.11980537   [answer]