91Ó°ÊÓ

The expected value, often symbolized as \( E(X) \), is a crucial concept in probability and statistics. It represents the average outcome you can anticipate from a random process over the long term. In simple terms, it is the mean of a probability distribution.
For instance, in our problem with phone calls at the university, you determine how many calls to expect in a given time frame. By knowing that calls come in at an average rate of one every two minutes, we first calculate how many calls there are per hour.

The calculation goes like this:

• In an hour, there are 60 minutes.
• Calls come at a rate of one call every two minutes, which gives us 30 two-minute intervals in an hour.

Thus, you can expect 30 calls in one hour. This expected value is important because it allows us to predict and manage resources effectively, such as staffing needed to handle the call volume.

Probability calculation involves determining the likelihood of certain events occurring within a given context. Using the Poisson distribution is especially suitable for finding the probability of a number of events within a fixed interval of time or space, particularly when these events happen with a known constant mean rate and independently of the time since the last event.

In our case, we are asked to calculate the probability of receiving three calls in a five-minute span. To do so, we use the Poisson probability formula:

• Set \( k = 3 \) for the number of calls we're interested in.
• Determine \( \lambda \), the expected average number of calls in five minutes, which we've computed as 2.5 calls.

The formula is:
\[ P(k; \lambda) = \frac{e^{-\lambda} \lambda^k}{k!} \]
Plugging in the values:\[ P(3; 2.5) = \frac{e^{-2.5} \cdot 2.5^3}{3!} \approx 0.2138 \]
This probability suggests there's about a 21% chance of exactly three calls occurring within any given five-minute segment.

The rate of occurrence refers to how frequently a certain event happens over a specific time period. It is often denoted as \( \lambda \) in the context of the Poisson distribution. Understanding this rate is essential as it forms the basis of further calculations like expected value or probability.

In our university phone registration example, we start with a rate of one call every two minutes. This translates first to a per-minute rate, \( \lambda = 0.5 \) calls per minute, which is an insightful measure. Once we know the rate per interval, we can easily scale this value across different time frames.
Consider a five-minute interval:

• Multiply the per-minute rate by five to find \( \lambda \) for this period.
• This results in 2.5 expected calls in five minutes.

This constant rate helps calculate probabilities over varying time spans and can aid significantly in planning and resource allocation, ensuring efficient management of call handling capacity.