Calculating Variance of a Linear Transformation of a Random Variable

In this article, we will explore how to calculate the variance of a linear transformation of a random variable. Specifically, we will calculate the variance of Y 2X - 5, given that the random variable X has a mean of 3 and a standard deviation of 5. We will use the properties of variance and standard deviation to find the solution and discuss the underlying mathematical principles.

Introduction to Variance and Standard Deviation

Variance is a measure of how spread out the values of a random variable are from its mean. It is defined as the average of the squared differences from the mean. The square root of the variance is the standard deviation, which provides a measure of the dispersion that is in the same units as the original variable.

Properties of Variance

There are two key properties of variance that we will use to solve this problem:

Variance Scaling

When a random variable is multiplied by a constant, the variance is multiplied by the square of that constant. If a is a constant and X is a random variable, then the variance of aX is given by:

Variance Scaling Property

Var(aX) a^2 · Var(X)

Constant Shift

Adding or subtracting a constant to a random variable does not affect its variance. If b is a constant, then the variance of X b (or X - b) is the same as the variance of X:

Constant Shift Property

Var(X b) Var(X) Var(X - b)

Step-by-Step Calculation

Given that the mean of X is 3 and the standard deviation is 5, we can start by calculating the variance of X:

Step 1: Calculate the variance of X

Since the standard deviation (SD) of X is 5, the variance (Var(X)) can be calculated as:

Var(X) (SD(X))^2 5^2 25

Step 2: Apply the Variance Scaling Property

We need to find the variance of Y 2X - 5. First, we apply the variance scaling property:

Var(2X) 2^2 · Var(X) 4 · 25 100

Step 3: Apply the Constant Shift Property

Next, we subtract 5 from the expression, but this does not change the variance:

Var(2X - 5) Var(2X) 100

Thus, the variance of Y 2X - 5 is 100.

Alternative Method Using Definitions

We can verify the solution using the definition of variance. We know that:

Var(aX b) E[(aX b - E[aX b])^2]

Using the linearity of expectation, we can rewrite the expression inside the expectation:

Var(2X - 5) E[2X - 5 - (2EX - 5)]^2 E[2X - 2EX]^2 4Var(X) 4 · 25 100

Conclusion and Further Insights

The variance of the linear transformation Y 2X - 5 is 100, given that the mean of X is 3 and the standard deviation is 5. This calculation demonstrates the application of variance scaling and constant shift properties, and provides a detailed step-by-step solution using both properties and the definition of variance.

Keywords

Variance, Linear Transformation, Standard Deviation