Understanding Variance in Linear Combinations: The Calculation of Var(3x - y) When SD of x is 1 and SD of y is 3

In statistical analysis, understanding the variance of a linear combination of random variables is a fundamental concept, especially when working with problems that involve scaling and shifting. This article will walk you through the process of calculating the variance of the expression 3x - y, given that the standard deviation (SD) of x is 1 and the SD of y is 3.

Prerequisites

To follow this article, it is essential to have a basic understanding of the following concepts:

Variance and Standard Deviation Linear Combinations of Random Variables The Properties of Variance for Linear Combinations of Independent Random Variables

The Formula and Its Application

The formula for the variance of a linear combination of two independent random variables X and Y is:

V[αX - βY] α2V[X] - β2V[Y] 2αβCov[X,Y]

If X and Y are independent, then their covariance is 0, simplifying the formula to:

V[αX - βY] α2V[X] β2V[Y]

Given Values

The standard deviation (SD) of x (SD(x)) is 1. The standard deviation (SD) of y (SD(y)) is 3.

Given this information, we can find the variances:

V[X] (SD(X))2 12 1 V[Y] (SD(Y))2 32 9

Calculation Steps

Since we are looking for V[3x - y], we have α 3 and β -1. Apply the formula:

V[3x - y]  (32 * V[X])   ((-1)2 * V[Y])
V[3x - y]  (9 * 1)   (1 * 9)
V[3x - y]  9   9
V[3x - y]  18

Thus, the variance of 3x - y is 18.

Conclusion

Understanding how to calculate the variance of linear combinations of random variables is crucial in statistical analysis and data science. Whether you are working with financial data, experimental measurements, or any other form of data, this knowledge can help you make more informed decisions and predictions.

For further exploration, consider studying related topics such as covariance, the properties of variance, and the application of these concepts in real-world scenarios.