1. Observe the number of customers visiting a particular ATM Centre during any day from March 21 to 26, 2011 between 10 a.m. – 12 noon with a time interval of 5 minutes (number of customers using the ATM every 5 minutes). Calculate the average number of customers arrived during 2 hour period.

Apply the probability and theoretical distribution concepts and interpret the results.
Also find the probability of:
a. No customer arrives in the given time interval;
b. More than two customers arrive in the given time interval.

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To calculate the average number of customers arrived during a 2-hour period at the ATM Centre, we need to first determine the number of customers visiting the ATM every 5 minutes from March 21 to 26, 2011, between 10 a.m. and 12 noon.

Given the time interval of 5 minutes, we can start by noting down the number of customers visiting the ATM at every 5-minute interval during the specified time period. Let's assume the number of customers for each 5-minute interval are as follows:

March 21, 2011:
10:00 a.m. - 0 customers
10:05 a.m. - 2 customers
10:10 a.m. - 1 customer
...
12:00 p.m. - 3 customers

Similarly, note down the number of customers for each 5-minute interval on the other days between March 21 and March 26, 2011.

Next, sum up the total number of customers across all the 5-minute intervals during the specified time period. In this case, sum up the numbers of customers for each of the intervals from March 21 to 26, 2011, between 10 a.m. and 12 noon.

Once you have the total number of customers, divide it by the total number of 5-minute intervals (this would be the number of days multiplied by the number of intervals per hour) to calculate the average number of customers arrived during the 2-hour period.

To apply probability and theoretical distribution concepts, we can use the average number of customers as the mean value for a Poisson distribution. The Poisson distribution is commonly used to model the number of events occurring in a fixed interval of time when the events are rare and independent of each other.

a. To find the probability of no customers arriving in the given time interval, simply calculate the Poisson probability of having 0 customers in the specified 2-hour interval using the average number of customers as the mean parameter.

b. To find the probability of more than two customers arriving in the given time interval, calculate the cumulative Poisson probability from 3 customers onwards (since more than 2 customers means at least 3 customers).

By applying the Poisson distribution, you can interpret the results and obtain the probabilities for the specified scenarios.