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Let's work through this problem step by step.

### a) Are the terms "mean" and "expected value" synonyms?

In probability theory and statistics, **yes**, the terms "mean" and "expected value" are synonyms. The **mean** of a probability distribution is the same as its **expected value**, representing the average outcome you would expect from a large number of trials.

So, the answer to part **a)** is: **Yes, they are synonyms.**

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### b) Find the expected value \( E(X) \) for the given probability density function.

The expected value of a continuous random variable \( X \) with probability density function (PDF) \( f(x) \) is given by:

\[
E(X) = \int_{a}^{b} x f(x) \, dx
\]

Here, \( f(x) = \frac{1}{24} x \), and the interval is \([4, 8]\).

Let's set up the integral:

\[
E(X) = \int_{4}^{8} x \cdot \frac{1}{24} x \, dx = \frac{1}{24} \int_{4}^{8} x^2 \, dx
\]

Now, calculate the integral:

\[
\int x^2 \, dx = \frac{x^3}{3}
\]

Evaluating this from 4 to 8:

\[
E(X) = \frac{1}{24} \left( \frac{8^3}{3} - \frac{4^3}{3} \right) = \frac{1}{24} \left( \frac{512}{3} - \frac{64}{3} \right)
\]

\[
E(X) = \frac{1}{24} \cdot \frac{448}{3} = \frac{448}{72} = \frac{112}{18} \approx 6.222
\]

So, the expected value \( E(X) \) is approximately **6.222**, rounded to three decimal places.

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Would you like further details on any step? Here are five related questions to deepen understanding:

1. What is the significance of the expected value in real-world applications?
2. How do you interpret the PDF \( f(x) = \frac{1}{24} x \) in this context?
3. How would you compute the variance for this probability density function?
4. What happens to the expected value if the interval changes from \([4, 8]\) to a wider range?
5. How does the expected value differ for discrete and continuous random variables?

**Tip**: When dealing with continuous probability distributions, always ensure the function integrates to 1 over the specified interval.

Gabriela was given the probability density function f(x) = 1/24 * x over the interval [4, 8]. In the context of probability distributions, are the terms 'mean' and 'expected value' synonyms? What is the expected value E(X)?

This problem explores the calculation of the expected value for a continuous random variable with the probability density function f(x) = 1/24 * x over the interval [4, 8]. Learn how to compute the expected value and understand the synonymity between 'mean' and 'expected value'.

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