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It appears that you're working on a problem related to finding the variance of a uniformly distributed random variable \( X \) over the interval \( (a, b) \), and you're tasked with verifying that:

\[
\text{Var}(X) = \frac{(b - a)^2}{12}
\]

Additionally, the expected value \( E(X) \) is given as \( E(X) = \frac{a + b}{2} \).

From the image, the task is to arrange the given expressions in the correct order to prove the formula for the variance. To solve this, let me help break down the process.

### Steps to prove Var(X):
1. **Formula for Variance**:  
   \[
   \text{Var}(X) = E\left(X^2\right) - \left(E(X)\right)^2
   \]
   First, we need to find both \( E(X^2) \) and \( \left(E(X)\right)^2 \).

2. **Find \( E(X) \)**:  
   This is already given as:
   \[
   E(X) = \frac{a + b}{2}
   \]
   Therefore, \( \left(E(X)\right)^2 = \left(\frac{a + b}{2}\right)^2 = \frac{(a + b)^2}{4} \).

3. **Find \( E(X^2) \)**:  
   For a uniform distribution, \( E(X^2) \) is computed using the integral:
   \[
   E(X^2) = \int_a^b x^2 \cdot \frac{1}{b - a} dx
   \]
   Evaluating this gives:
   \[
   E(X^2) = \frac{b^3 - a^3}{3(b - a)}
   \]

4. **Compute Variance**:  
   Now, using the formula for variance:
   \[
   \text{Var}(X) = E(X^2) - \left(E(X)\right)^2
   \]
   Substitute the values:
   \[
   \text{Var}(X) = \frac{b^3 - a^3}{3(b - a)} - \frac{(a + b)^2}{4}
   \]

5. **Simplify**:  
   Simplifying the expression eventually leads to:
   \[
   \text{Var}(X) = \frac{(b - a)^2}{12}
   \]

---

### Now, based on the boxes provided:
- You need to arrange the expressions such that they show the calculation of variance step-by-step, starting from the given formula for \( \text{Var}(X) \).

Would you like me to walk through the exact placement of the boxes? Or do you have a particular part you want clarified further?

If X is a uniform distribution defined over the interval (a, b), verify the following formula. You will use the fact that E(X) = (a + b) / 2 and Var(X) = (b - a)^2 / 12.

Learn how to verify the variance formula Var(X) = (b - a)^2 / 12 for a uniformly distributed random variable X over the interval (a, b). This problem covers key concepts in probability theory and statistics, including expected value and variance calculations.

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