Question

In a reality show, two judges independently provided marks based on the performance of the participants. If the marks provided by the second judge are given by \( Y = 10.5 + 2X \), where \( X \) is the marks provided by the first judge. If the variance of the marks provided by the second judge is 100, then the variance of the marks provided by the first judge is:

• A. 50
• B. 25
• C. 99
• D. 49.5

Hint:

When dealing with linear transformations, use the formula \( {Var}(Y) = a^2 {Var}(X) \) to calculate the variance of the transformed variable.

Answer 1

The Correct Option is B

Step 1: Understand the relationship between the marks. The marks provided by the second judge, \( Y \), are related to the marks provided by the first judge, \( X \), by the equation: \[ Y = 10.5 + 2X. \] This is a linear transformation of \( X \), where the variance of \( Y \) is related to the variance of \( X \). 
Step 2: Use the variance transformation formula. For a linear transformation \( Y = aX + b \), the variance of \( Y \) is given by: \[ {Var}(Y) = a^2 {Var}(X). \] Here, \( a = 2 \), so: \[ {Var}(Y) = 2^2 {Var}(X) = 4 {Var}(X). \] We are given that \( {Var}(Y) = 100 \), so: \[ 100 = 4 {Var}(X) \quad \Rightarrow \quad {Var}(X) = \frac{100}{4} = 25. \] Thus, the correct answer is: \[ \boxed{25}. \]