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 mark provided by the first judge, and the variance of the marks provided by the second judge is 100, what is the variance of the marks provided by the first judge?
This problem is solved by applying a fundamental property of variance that describes how it behaves under linear transformations.
If a variable X is transformed into a new variable Y using the equation Y = aX + b (where 'a' and 'b' are constants), their variances are related by the following formula:
Var(Y) = a² * Var(X)
Note: Adding a constant (the 'b' value) shifts the entire dataset but doesn't change its spread, which is why 'b' does not affect the final variance. Multiplying by a constant 'a' scales the spread, and its effect on variance is squared.
We are given the equation Y = 10.5 + 2X. Comparing this to the general form Y = b + aX, we can identify our constants:
a = 2b = 10.5We substitute the value of 'a' into the property formula:
Var(Y) = 2² * Var(X) Var(Y) = 4 * Var(X) We are given that the variance of Y is 100. We can now plug this value into our equation and solve for the unknown variance of X.
100 = 4 * Var(X) Var(X) = 100 / 4 Var(X) = 25 Final Answer: The variance of the marks provided by the first judge is 25.
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