https://file.aiquickdraw.com/mathbot/file/1726772623187amlf977p.jpg

Let's break down the question and analyze the statements based on the rules of variance:

1. **The variance of \( X + 1 \)**:  
   When adding or subtracting a constant to a random variable, the variance remains unchanged. This is because the variance only measures the spread of the data and shifting it does not affect the spread. Therefore, the variance of \( X + 1 \) is the same as the variance of \( X \), which is 2.  
   **This statement is true.**

2. **The variance of \( -2X \)**:  
   When multiplying a random variable by a constant \( c \), the variance is scaled by \( c^2 \). So, the variance of \( -2X \) is given by:
   \[
   \text{Var}(-2X) = (-2)^2 \times \text{Var}(X) = 4 \times 2 = 8
   \]
   This means the variance of \( -2X \) is 8, not -4.  
   **This statement is false.**

### Final Answer:
- The first statement is **true**.
- The second statement is **false**.

Would you like a further breakdown or more details?

Here are five related questions to deepen your understanding:
1. What happens to the variance if you multiply a random variable by a constant?
2. How does adding a constant affect the mean of a random variable?
3. What is the standard deviation of \( X + 1 \) if the standard deviation of \( X \) is known?
4. Can the variance of a random variable ever be negative?
5. How do transformations like \( aX + b \) impact both the variance and mean of a random variable?

**Tip:** Variance is always non-negative, and it scales by the square of a constant when the random variable is multiplied by that constant.

If the variance of a random variable X is 2, which of the following can be said to be always true? The variance of X + 1 is 2. The variance of -2X is -4.

Explore the effects on variance when a constant is added or when a random variable is scaled. If the variance of X is 2, does the variance change when transformed into X + 1 or -2X? Learn how key properties of variance apply in this example, suitable for undergraduate students.

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