A person works on a help line for a company.  They spend most of their time on the phone but in any hour of work there may be time spent not on the phone, eg making notes, attending to email etc.

Let X = the number of phone calls that the person takes in 1 hour and assume X ~ Poi(6).  Furthermore let Y = time taken to deal with a call (in mins) and assume Y ~ N(8,2.52)

Based on the above information, what is the average amount of time spent on the phone in 1 hour of work?

a.

6 *8 = 48 minutes

b.

impossible to calculate without more information

c.

6 * 8 + 1.96 * 2.5 = 52.9 minutes

d.

6 * 8 + 2.52 = 54.25 minutes

To calculate the **average amount of time** spent on the phone in 1 hour, we need to consider both the number of phone calls (which follows a Poisson distribution) and the average time taken per call (which follows a normal distribution).

### Given:
- \( X \sim \text{Poisson}(6) \) : This means the expected number of phone calls in an hour is \( \mathbb{E}(X) = 6 \).
- \( Y \sim \text{Normal}(8, 2.5^2) \) : This means the average time per phone call is \( \mathbb{E}(Y) = 8 \) minutes, with a standard deviation of 2.5 minutes.

### Approach:
The total time spent on the phone in one hour is the product of:
- The number of calls, \( X \), in that hour.
- The time spent on each call, \( Y \).

Thus, the **expected time spent on the phone in one hour** is:

\[
\mathbb{E}(X \cdot Y) = \mathbb{E}(X) \cdot \mathbb{E}(Y)
\]

Since \( \mathbb{E}(X) = 6 \) and \( \mathbb{E}(Y) = 8 \), we get:

\[
\mathbb{E}(X \cdot Y) = 6 \cdot 8 = 48 \text{ minutes}.
\]

### Conclusion:
The average time spent on the phone in 1 hour is **48 minutes**, which corresponds to option **a**.

### Answer: 
**a. 6 * 8 = 48 minutes**

Would you like more details or clarification? Here are five related questions to expand on this:

1. How would the total time on the phone change if the number of calls followed a different distribution?
2. What is the variance of the total time spent on the phone in one hour?
3. How does the standard deviation of time spent per call affect the total time?
4. How could you calculate the probability of spending less than 40 minutes on the phone in one hour?
5. How does the assumption that calls are independent impact this calculation?

**Tip**: The expected value (mean) of a product of two independent random variables is the product of their expected values!

A person works on a help line for a company. They spend most of their time on the phone but in any hour of work there may be time spent not on the phone, e.g., making notes, attending to email etc. Let X = the number of phone calls that the person takes in 1 hour and assume X ~ Poi(6). Furthermore let Y = time taken to deal with a call (in mins) and assume Y ~ N(8, 2.52). Based on the above information, what is the average amount of time spent on the phone in 1 hour of work? a. 6 * 8 = 48 minutes b. impossible to calculate without more information c. 6 * 8 + 1.96 * 2.5 = 52.9 minutes d. 6 * 8 + 2.52 = 54.25 minutes

Learn how to calculate the average time spent on phone calls in one hour using Poisson and Normal distributions. This detailed solution uses expected value calculations based on X ~ Poi(6) and Y ~ N(8, 2.5^2), providing a clear step-by-step approach. Ideal for undergraduate students studying probability theory.

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