In order to determine the variance of the random variable \(y\) defined as \(y = 2x + 5\), we must understand the properties of variance in relation to linear transformations of random variables.
Understanding Variance
- Variance Definition: Variance is a measure of how much values in a dataset differ from the mean. It is denoted as \(Var(x)\).
- Given: The variance of the random variable \(x\) is \(Var(x) = 18\).
Linear Transformation of Random Variables
- Transformation Rule: For a linear transformation of the form \(y = ax + b\):
- The variance of \(y\) is calculated using the formula:
\[ Var(y) = a^2 \cdot Var(x) \]
- Here, \(a\) is the coefficient of \(x\) and \(b\) is a constant.
Applying the Transformation Rule
- Identifying Values: In this case, \(a = 2\) and \(b = 5\).
- Calculating Variance:
- Substitute \(a\) and \(Var(x)\) into the formula:
\[ Var(y) = 2^2 \cdot Var(x) \]
\[ Var(y) = 4 \cdot 18 \]
\[ Var(y) = 72 \]
Conclusion
- The variance of the random variable \(y\) is:
\[ Var(y) = 72 \]
- Key Insight: The constant \(b\) does not affect the variance, while the coefficient \(a\) scales the variance by the square of its value.
This understanding of variance and its properties is essential for analyzing random variables, especially in statistics and probability theory.