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Let's work through each part of the question.

### Given:
- \( E(X) = 26.2 \) (Expected value of \( X \))
- \( \text{Var}(X) = 27 \) (Variance of \( X \))

#### Part (a): 
For \( A = 3X \), we need to find the expected value \( E(A) \) and the variance \( \text{Var}(A) \).

1. **Expected value**:
   \[
   E(A) = E(3X) = 3 \cdot E(X) = 3 \cdot 26.2 = 78.6
   \]

2. **Variance**:
   \[
   \text{Var}(A) = \text{Var}(3X) = 3^2 \cdot \text{Var}(X) = 9 \cdot 27 = 243
   \]

#### Part (b): 
For \( B = 3X - 3 \), we need to find \( E(B) \) and \( \text{Var}(B) \).

1. **Expected value**:
   \[
   E(B) = E(3X - 3) = 3 \cdot E(X) - 3 = 3 \cdot 26.2 - 3 = 78.6 - 3 = 75.6
   \]

2. **Variance**:
   \[
   \text{Var}(B) = \text{Var}(3X - 3) = \text{Var}(3X) = 9 \cdot 27 = 243
   \]
   (Since subtracting a constant does not affect variance.)

#### Part (c): 
For \( C = -3X + 8 \), we need to find \( E(C) \) and the standard deviation \( \text{SD}(C) \).

1. **Expected value**:
   \[
   E(C) = E(-3X + 8) = -3 \cdot E(X) + 8 = -3 \cdot 26.2 + 8 = -78.6 + 8 = -70.6
   \]

2. **Variance**:
   \[
   \text{Var}(C) = \text{Var}(-3X + 8) = (-3)^2 \cdot \text{Var}(X) = 9 \cdot 27 = 243
   \]
   (Since adding a constant does not affect variance.)

3. **Standard deviation**:
   \[
   \text{SD}(C) = \sqrt{\text{Var}(C)} = \sqrt{243} \approx 15.59
   \]

---

### Summary of Results:
- **Part (a)**:
  - \( E(A) = 78.6 \)
  - \( \text{Var}(A) = 243 \)
  
- **Part (b)**:
  - \( E(B) = 75.6 \)
  - \( \text{Var}(B) = 243 \)

- **Part (c)**:
  - \( E(C) = -70.6 \)
  - \( \text{SD}(C) = 15.59 \)

Would you like more details or have any questions?

### Related Questions:
1. How does multiplying a random variable by a constant affect its variance?
2. What happens to the expected value when a constant is added to a random variable?
3. How does variance behave when we subtract a constant from a random variable?
4. What is the relationship between standard deviation and variance?
5. Can we compute the covariance between \( A \), \( B \), and \( C \)?

#### Tip: 
The standard deviation is always the square root of the variance, so changing the variance will always impact the spread (or dispersion) of the data.

X is a random variable with mean/expected value E(X) = 26.2 and variance VAR(X) = 27. Consider the following random variables. A = 3X, B = 3X - 3, C = -3X + 8. Answer parts (a) through (c).

Solve a problem involving expected value, variance, and standard deviation for random variables. The problem explores transformations of random variables and their effects on mean and variance, with detailed calculations for random variables A, B, and C. Suitable for college-level students studying probability and statistics.

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