The concept of linear combinations of normally distributed variables is a significant topic in probability and statistics. This section delves into how these combinations maintain their normal distribution, an essential aspect of understanding real-world data distributions.

Understanding Normal Distribution

• Normal Distribution: A bell-shaped curve described by its mean $μ$ and variance $σ²$, used to model various real-world phenomena.

Linear Combinations in Normal Distribution

• Linear Combination: The process of adding variables together, each multiplied by a constant. It's a foundation in statistics for modeling and analysis.

Key Principle

• Linear Combinations of Normally Distributed Variables: If the variables are normally distributed, their linear combinations also follow a normal distribution.
• Single Variable: If $X \sim N(\mu, \sigma^2)$, then $aX + b$ is also normally distributed as $N(a\mu + b, a^2\sigma^2$.
• Two Independent Variables: For $X \sim N(\mu_1, \sigma_1^2)$ and $Y \sim N(\mu_2, \sigma_2^2)$, $aX + bY$ follows $N(a\mu_1 + b\mu_2, a^2\sigma_1^2 + b^2\sigma_2^2)$.

Examples

Example 1: Single Variable Transformation

Question: Given a normal distribution $N(10, 4)$ for a random variable $X$, determine the distribution of the transformed variable $3X + 2$.

Solution:

1. Mean of (3X + 2):

Calculated using $E(3X + 2) = 3E(X) + 2$.

With $E(X) = 10$, it results in $E(3X + 2) = 3 \times 10 + 2 = 32$.

2. Variance of (3X + 2):

Determined by $\text{Var}(3X + 2) = 3^2 \text{Var}(X)$.

Given $\text{Var}(X) = 4$, this leads to $\text{Var}(3X + 2) = 9 \times 4 = 36$.

Conclusion: The transformed variable $3X + 2$ follows a normal distribution $N(32, 36)$.

Example 2: Combination of Two Independent Variables

Question: What is the distribution of the variable $2X - Y$ if $X$ and $Y$ are independent normally distributed variables with $X \sim N(5, 9)$ and $Y \sim N(3, 16)$?

Solution:

1. Mean: The mean of $2X - Y$ is $2 \times 5 - 3 = 7$.

2. Variance: The variance of $2X - Y$ is $2^2 \times 9 + (-1)^2 \times 16 = 52.$

Conclusion: $2X - Y$ is normally distributed with $N(7, 52)$.