To determine the relationship between the variances of two continuous random variables, X and Y, given the linear relationship \(Y = 2X + 3\), we need to apply the fundamental properties of variance.

Variance of Random Variables

In probability theory and statistics, the variance of a random variable is a measure of how much the values of the random variable differ from its expected value (mean). A high variance indicates that data points are generally far from the mean, while a low variance indicates that data points are generally close to the mean.

• For a random variable X, its variance is denoted as \(\text{Var}(X)\) or \(\sigma_X^2\).
• It is formally defined as \( \text{Var}(X) = E[(X - E[X])^2] \), where \(E[X]\) is the expected value (mean) of X.

Properties of Variance

When dealing with linear transformations of random variables, specific properties of variance come into play. For any random variable X and any constants 'a' and 'b', the variance of the linear transformation \(aX + b\) is given by the following property:

• The variance of a constant multiplied by a random variable is the square of the constant times the variance of the random variable: \[ \text{Var}(aX) = a^2 \text{Var}(X) \]
• The variance of a random variable plus a constant is simply the variance of the random variable, as adding a constant shifts the distribution but does not change its spread: \[ \text{Var}(X + b) = \text{Var}(X) \]
• Combining these two properties, for a linear transformation of the form \(Y = aX + b\), the variance of Y is: \[ \text{Var}(Y) = \text{Var}(aX + b) = a^2 \text{Var}(X) \]

Applying Variance Properties to \(Y = 2X + 3\)

Given the relationship between the continuous random variables X and Y as \(Y = 2X + 3\), we can directly apply the variance property for linear transformations.

In our given relationship \(Y = 2X + 3\):

• The constant 'a' (the coefficient of X) is \(2\).
• The constant 'b' (the additive constant) is \(3\).

Using the property \( \text{Var}(Y) = a^2 \text{Var}(X) \):

1. Substitute the value of 'a' into the formula: \[ \text{Var}(Y) = (2)^2 \text{Var}(X) \]
2. Calculate the square of 'a': \[ \text{Var}(Y) = 4 \text{Var}(X) \]

Since \(\sigma_Y^2\) denotes the variance of Y and \(\sigma_X^2\) denotes the variance of X, we can write the relationship as:

\[ \sigma_Y^2 = 4 \sigma_X^2 \]

Conclusion on Variance Relationship

Based on the properties of variance for linear transformations of random variables, the variance of Y, \(\sigma_Y^2\), is found to be four times the variance of X, \(\sigma_X^2\). It is important to note that the additive constant \(+3\) in the relationship \(Y = 2X + 3\) does not affect the variance. This is because adding a constant to a random variable only shifts its mean, but it does not change the spread or variability of its distribution. Therefore, the variance remains unaffected by the additive constant.

Thus, the correct relationship between the variances is:

\[ \sigma_Y^2 = 4 \sigma_X^2 \]