https://file.aiquickdraw.com/mathbot/file/1727501178023tgen3t64.jpg

The problem from the image involves using an Erlang distribution for customer arrivals modeled as an exponential random variable with a mean of \(\mu = 5\) minutes. There are two parts of the problem:

**Part (a)** asks for the expected value of the time by which the first 6 customers have arrived. The expected value for an Erlang distribution (which is a sum of exponential distributions) can be calculated using the formula:
\[
E(Y) = k \cdot \mu
\]
where \(k\) is the number of customers, and \(\mu\) is the mean arrival time.

In this case:
- \(k = 6\) (since we're dealing with the first 6 customers)
- \(\mu = 5\) minutes.

Thus:
\[
E(Y) = 6 \cdot 5 = 30 \text{ minutes}.
\]

So, the expected time for the first 6 customers to arrive is **30 minutes**.

---

**Part (b)** asks for the probability that the first 6 customers arrive within 15 minutes or sooner. For an Erlang distribution, the probability can be found using the cumulative distribution function (CDF) of the Erlang distribution. The CDF for an Erlang distribution is given by:
\[
P(Y \leq t) = 1 - \sum_{n=0}^{k-1} \frac{(t/\mu)^n e^{-t/\mu}}{n!}
\]
where:
- \(t = 15\) minutes (the time within which we want the arrivals),
- \(\mu = 5\) minutes (mean),
- \(k = 6\) (number of customers).

I can calculate this for you now.The probability that the first 6 customers arrive within 15 minutes or sooner is approximately **0.0839**, or about **8.39%**.

Let me know if you'd like more details or if you have any questions.

Here are 5 related questions to deepen understanding:
1. How does the Erlang distribution differ from the exponential distribution?
2. What would change if the mean arrival time were reduced to 3 minutes instead of 5?
3. How is the Erlang distribution related to the Gamma distribution?
4. How can we compute the variance of the arrival time for the first 6 customers?
5. What are some real-world applications of the Erlang distribution beyond customer arrivals?

**Tip**: The Erlang distribution is especially useful when modeling the sum of several exponential random variables with the same rate.

Customers arrive at an automatic teller machine at time intervals modeled as an exponential random variable with mean μ = 5 minutes. (a) What is the expected time by which the first 6 customers have arrived? (b) What is the probability that the first 6 customers arrive within 15 minutes or sooner?

Solve a problem involving customer arrivals at an ATM using an Erlang distribution. Calculate the expected time for 6 customers to arrive and the probability that they arrive within 15 minutes. Key concepts include the Erlang distribution and its cumulative distribution function.

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