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.