Question

Two continuous random variables \( X \) and \( Y \) are related as
\( Y = 2X + 3 \).
Let \( \sigma_X^2 \) and \( \sigma_Y^2 \) denote the variances of \( X \) and \( Y \), respectively. The variances are related as

• A. \( \sigma_Y^2 = 2 \sigma_X^2 \)
• B. \( \sigma_Y^2 = 4 \sigma_X^2 \)
• C. \( \sigma_Y^2 = 5 \sigma_X^2 \)
• D. \( \sigma_Y^2 = 25 \sigma_X^2 \)

Hint:

For a linear transformation of a random variable \( Y = aX + b \), the variance of \( Y \) is \( \sigma_Y^2 = a^2 \sigma_X^2 \).

Answer 1

The Correct Option is B

The relationship between \( X \) and \( Y \) is given by \( Y = 2X + 3 \). The variance of \( Y \) is related to the variance of \( X \) as follows: \[ \sigma_Y^2 = \text{Var}(2X + 3). \] Since the variance of a constant is zero, we have: \[ \sigma_Y^2 = 4 \, \text{Var}(X) = 4 \sigma_X^2. \] Final Answer: \( \sigma_Y^2 = 4 \sigma_X^2 \)