Suppose that calls at a home security monitoring company are receive according t

Question

Suppose that calls at a home security monitoring company are receive according to a Poisson process with an average of 1.4 calls per hour. Ted’s job is to monitor the calls for an 8-hour shift.

Let X= the number of calls that Ted receives during an 8-hour shift.

-Is X discrete or continuous?

-What is the expected (average) number of calls that Ted receives in an 8-hour shift?

-What is the probability that Ted will receive fewer than the expected number of calls in an 8-hour shift?

Let Y= the time from the beginning of his shift until Ted receives the first call.

-Is Y discrete or continuous?

-How long should Ted expect to wait until the first call arrives? Give your answer in hours.

-It takes Ted 15 minutes to drink his coffee. What is the probability that the first call will be received after Ted finishes drinking his coffee?

If Ted receives more than 16 calls during his shift, his boss gives him a gift card to the local coffee shop.

-Show how you could use the Poisson distribution to find the probability that Ted will earn a gift card during his next 8-hour shift. Make sure to show the details of your work.

-Show how you could use the Gamma distribution to find the probability that Ted will earn a gift card during his next 8-hour shift. Make sure to show the details of your work. Set up an integral with the integrand and appropriate limits. Then you can use a definite integral calculator to get your final answer. (Unless you like doing integration by parts 15 times…)

Explanation / Answer

Suppose that calls at a home security monitoring company are receive according to a Poisson process with an average of 1.4 calls per hour. Ted’s job is to monitor the calls for an 8-hour shift.

Let X= the number of calls that Ted receives during an 8-hour shift.

-Is X discrete or continuous?

DISCRETE, as it only takes integer values.

*************************

-What is the expected (average) number of calls that Ted receives in an 8-hour shift?

He gets 1.4 calls/hr, so that 1.4*8 = 11.2 calls. [ANSWER]

***************************

-What is the probability that Ted will receive fewer than the expected number of calls in an 8-hour shift?

That means he gets at most 11 calls.

Using a cumulative poisson distribution table or technology, matching          
          
u = the mean number of successes =    11.2      
          
x = the maximum number of successes =    11      
          
Then the cumulative probability is          
          
P(at most   11   ) =    0.555405483 [ANSWER]

*****************************

Let Y= the time from the beginning of his shift until Ted receives the first call.

-Is Y discrete or continuous?

CONTINUOUS, as it takes on any positive value.

******************************

-How long should Ted expect to wait until the first call arrives? Give your answer in hours.

There are 1.4 calls/hr, so he expects the first call to be in 1/1.4 = 0.714285714 hours. [ANSWER]

*****************************

-It takes Ted 15 minutes to drink his coffee. What is the probability that the first call will be received after Ted finishes drinking his coffee?

Note that 15 min = 0.25 hr.

mean = u =    0.714285714      

The right tailed area in an exponential distribution is          
          
P(x>x1) = e^(-x1/u)          
          
As          
          
x1 = critical value =    0.25      
          
          
Then          
          
P =    0.70468809   [ANSWER]  

*******************************************

Hi! Please submit the next part as a separate question. That way we can continue helping you! Please indicate which parts are not yet solved when you submit. Thanks!

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.