Poisson distribution

Telephone calls arrive at a call centre randomly, at an average rate of 1.7 per minute. After the call centre was closed for a week, in a random sample of 10 minutes there were 25 calls to the call centre.

Carry out a suitable test to determine whether or not there is evidence that the rate of calls arriving at the call centre has changed. Use a 5% level of significance and state your hypotheses clearly.

Only 1.2% of the calls to the call centre last longer than 8 minutes.

One day Tiang has 70 calls.

Find the probability that out of these 70 calls Tiang has more than 2 calls lasting longer than 8 minutes.

The call centre records show that 95% of days have at least one call lasting longer than 30 minutes. On Wednesday 900 calls arrived at the call centre and none of them lasted longer than 30 minutes.

Use a Poisson approximation to estimate the proportion of calls arriving at the call centre that last longer than 30 minutes.

During the summer, mountain rescue team A receives calls for help randomly with a rate of 0.4 per day.

Find the probability that during the summer, mountain rescue team A receives at least 19 calls for help in 28 randomly selected days.

The leader of mountain rescue team A randomly selects 250 summer days from the last few years. She records the number of calls for help received on each of these days.

Using a Poisson approximation, estimate the probability of the leader finding at least 20 of these days when more than 1 call for help was received by mountain rescue team A.

Mountain rescue team A believes that the number of calls for help per day is lower in the winter than in the summer. The number of calls for help received in 42 randomly selected winter days is 8

Use a suitable test, at the 5% level of significance, to assess whether or not there is evidence that the number of calls for help per day is lower in the winter than in the summer. State your hypotheses clearly.

During the summer, mountain rescue team B receives calls for help randomly with a rate of 0.2 per day, independently of calls to mountain rescue team A.

The random variable C is the total number of calls for help received by mountain rescue teams A and B during a period of n days in the summer. On a Monday in the summer, mountain rescue teams A and B each receive a call for help.

Given that over the next n days P(C = 0) < 0.001

calculate the minimum value of n

Write down an assumption that needs to be made for the model to be appropriate.

On a weekday, a garage receives telephone calls randomly, at a mean rate of 1.25 per 10 minutes.

Show that the probability that on a weekday at least 2 calls are received by the garage in a 30‑minute period is 0.888 to 3 decimal places.

Calculate the probability that at least 2 calls are received by the garage in fewer than 4 out of 6 randomly selected, non‑overlapping 30‑minute periods on a weekday.

The manager of the garage randomly selects 150 non‑overlapping 30‑minute periods on weekdays. She records the number of calls received in each of these 30‑minute periods.

Using a Poisson approximation show that the probability of the manager finding at least 3 of these 30‑minute periods when exactly 8 calls are received by the garage is 0.664 to 3 significant figures.

Explain why the Poisson approximation may be reasonable in this case.

The manager of the garage decides to test whether the number of calls received on a Saturday is different from the number of calls received on a weekday. She selects a Saturday at random and records the number of telephone calls received by the garage in the first 4 hours.

Write down the hypotheses for this test.

The manager found that there had been 40 telephone calls received by the garage in the first 4 hours.

Carry out the test using a 5% level of significance.

Members of a photographic group may enter a maximum of 5 photographs into a members only competition. Past experience has shown that the number of photographs, N, entered by a member follows the probability distribution shown below.

show that Var(N) = 2.76

The group decided to charge a 50p entry fee for the first photograph entered and then 20p for each extra photograph entered into the competition up to a maximum of £1 per person. Thus a member who enters 3 photographs pays 90p and a member who enters 4 or 5 photographs just pays £1

Assuming that the probability distribution for the number of photographs entered by a member is unchanged,

calculate the expected entry fee per member.

Bai suggests that, as the mean and variance are close, a Poisson distribution could be used to model the number of photographs entered by a member next year.

State a limitation of the Poisson distribution in this case.

The number of customers entering Jeff’s supermarket each morning follows a Poisson distribution.

Past information shows that customers enter at an average rate of 2 every 5 minutes.

Using this information,

find the probability that exactly 26 customers enter Jeff’s supermarket during a randomly selected 1-hour period one morning,

find the probability that at least 21 customers enter Jeff’s supermarket during a randomly selected 1-hour period one morning.

A rival supermarket is opened nearby. Following its opening, the number of customers entering Jeff’s supermarket over a randomly selected 40-minute period is found to be 10

Test, at the 5% significance level, whether or not there is evidence of a decrease in the rate of customers entering Jeff’s supermarket. State your hypotheses clearly.

A further randomly selected 20-minute period is observed and the hypothesis test is repeated. Given that the true rate of customers entering Jeff’s supermarket is now 1 every 5 minutes,

calculate the probability of a Type II error.

The discrete random variables W, X and Y are distributed as follows

W ~ B(10, 0.4) X ~ Po(4) Y ~ Po(3)

Explain whether or not Po(4) would be a good approximation to B(10, 0.4)

State the assumption required for X + Y to be distributed as Po(7)

Given the assumption in part (b) holds,

find P(X + Y < Var(W))

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