Linear combinations of any normal random variables

Linear Combinations of Normal Random Variables Overview

A linear combination of normal random variables is itself a normal random variable. This fact is a key property of the normal distribution and a cornerstone of many advanced statistical techniques.
The simple form of a linear combination of random variables is given by Y = aX + bZ, where X and Z are the random variables, and a and b are constants.
In the context of Normal distributions, if X and Z are both normally distributed, Y is also normally distributed.

In a linear combination of normal random variables, the expectation of Y is given by E(Y) = aE(X) + bE(Z), where E() denotes the expected value.
The variance of Y in these circumstances is given by Var(Y) = a²Var(X) + b²Var(Z) + 2abCov(X,Z), where Var() signifies the variance and Cov() refers to covariance.

The Central Limit Theorem indicates that the sum of a large number of independent and identically distributed random variables, irrespective of their shape, will approximately follow a normal distribution. This concept is central to the properties of normal random variables and their linear combinations.

The concept of linear combinations of normal random variables is frequently utilised in regression analysis, to represent the result as a combination of predictor variables.
Experimentally, in disciplines such as Physics and Engineering, different measurements often generate variables that are summed or subtracted to produce a result.

An important condition for the linear combinations of normal random variables to be normal is that the variables involved should be independent. If this assumption is violated, the resulting distribution may not be normal.
Linear combinations assume the relationship between variables is primarily linear. If this assumption is not met, or if the relationship is non-linear, the utility of linear combinations tend to decrease.

If X ~ N(μ₁, σ₁²) and Y ~ N(μ₂, σ₂²) are independent normal variables, then X + Y and X - Y are also normally distributed.
The mean of X + Y is μ₁ + μ₂ and variance is σ₁² + σ₂². The mean of X - Y is μ₁ - μ₂ and variance is σ₁² + σ₂².