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The problem provides a probability distribution for the random variable \( x \) with the following values:

\[
\begin{array}{c|c}
x & f(x) \\
\hline
3 & 0.25 \\
6 & 0.50 \\
9 & 0.25 \\
\end{array}
\]

We need to calculate:

1. \( E(x) \) - The expected value of \( x \)
2. \( \sigma^2 \) - The variance of \( x \) (to 1 decimal place)
3. \( \sigma \) - The standard deviation of \( x \) (to 2 decimal places)

---

### Step (a): Expected Value \( E(x) \)

The expected value \( E(x) \) is given by the formula:

\[
E(x) = \sum (x_i \cdot f(x_i))
\]

Using the provided values of \( x \) and \( f(x) \):

\[
E(x) = (3 \cdot 0.25) + (6 \cdot 0.50) + (9 \cdot 0.25)
\]

\[
E(x) = 0.75 + 3.00 + 2.25 = 6.00
\]

Thus, the expected value \( E(x) = 6 \).

---

### Step (b): Variance \( \sigma^2 \)

The variance \( \sigma^2 \) is calculated as:

\[
\sigma^2 = \sum \left[ (x_i - E(x))^2 \cdot f(x_i) \right]
\]

Substituting \( E(x) = 6 \):

\[
\sigma^2 = (3 - 6)^2 \cdot 0.25 + (6 - 6)^2 \cdot 0.50 + (9 - 6)^2 \cdot 0.25
\]

\[
\sigma^2 = (-3)^2 \cdot 0.25 + (0)^2 \cdot 0.50 + (3)^2 \cdot 0.25
\]

\[
\sigma^2 = 9 \cdot 0.25 + 0 \cdot 0.50 + 9 \cdot 0.25 = 2.25 + 0 + 2.25 = 4.5
\]

Thus, the variance \( \sigma^2 = 4.5 \).

---

### Step (c): Standard Deviation \( \sigma \)

The standard deviation \( \sigma \) is the square root of the variance:

\[
\sigma = \sqrt{\sigma^2} = \sqrt{4.5} \approx 2.12
\]

Thus, the standard deviation \( \sigma \approx 2.12 \) (rounded to 2 decimal places).

---

Would you like more details on any of these steps or have any further questions?

Here are 5 related questions to explore:
1. How does changing the values of \( f(x) \) affect the expected value?
2. Can the expected value be a decimal even if \( x \) values are integers?
3. What happens to the variance if all \( f(x) \) values are the same?
4. Why is the standard deviation always a non-negative number?
5. How do the variance and standard deviation relate to the spread of a distribution?

**Tip:** The standard deviation is a more intuitive measure of spread because it has the same units as the original data.

The following table provides a probability distribution for the random variable x. Compute E(x), the expected value of x. Compute σ^2, the variance of x (to 1 decimal). Compute σ, the standard deviation of x (to 2 decimals).

Learn how to compute the expected value, variance, and standard deviation for a random variable x with a given probability distribution. This example uses the values x = {3, 6, 9} with probabilities f(x) = {0.25, 0.50, 0.25}. Detailed step-by-step solutions provided, suitable for Grades 11-12.

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