Math Problem Statement

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.

Solution

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

  1. The variance of X+1X + 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+1X + 1 is the same as the variance of XX, which is 2.
    This statement is true.

  2. The variance of 2X-2X:
    When multiplying a random variable by a constant cc, the variance is scaled by c2c^2. So, the variance of 2X-2X is given by: Var(2X)=(2)2×Var(X)=4×2=8\text{Var}(-2X) = (-2)^2 \times \text{Var}(X) = 4 \times 2 = 8 This means the variance of 2X-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+1X + 1 if the standard deviation of XX is known?
  4. Can the variance of a random variable ever be negative?
  5. How do transformations like aX+baX + 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.

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Math Problem Analysis

Mathematical Concepts

Statistics
Variance
Random Variables

Formulas

Var(X + c) = Var(X)
Var(aX) = a^2 * Var(X)

Theorems

Scaling property of variance
Addition of constants does not affect variance

Suitable Grade Level

Undergraduate level statistics