# Linear combinations of any normal random variables

## Linear combinations of any normal random variables

## Basic Definition

- A
**linear combination**of random variables refers to any sum of the random variables, each scaled by a non-random constant. - This concept can be applied to both
**discrete and continuous random variables**. - The constants are deterministic and can be adjusted based on specific conditions or scenario requirements.

## Linear Combinations and Expected Values

- For two random variables X and Y, and any two constants a and b, the
**expected value**of the linear combination is expressed as E(aX + bY) = aE(X) + bE(Y). - The expected value presents a central tendency measure, providing a sense of the distribution’s “centre”.
- Utilising this formula can simplify calculations, especially within the realms of statistics and probability.

## Linear Combinations and Variances

- The
**variance**of a linear combination is formulated as Var(aX + bY) = a^2 Var(X) + b^2 Var(Y) + 2abCov(X,Y), where X and Y are not independent. - If X and Y are
**independent variables**, the variance formula simplifies to Var(aX + bY) = a^2 Var(X) + b^2 Var(Y), resulting from Cov(X,Y) equalling 0. - Variance offers a measure of how much the random variable values disperse around the mean.

## Distributions of Linear Combinations

- The distribution of a linear combination of two independent random variables can be defined by their
**convolution**. - The distribution function of the variables’ linear combination can be obtained via the convolution of each variable’s individual distribution function.
- Specifically, in cases where the random variables are normally distributed, their linear combination is also
**normally distributed**.

## Key Properties

- Linear combinations of random variables retain the properties of
**Linearity of Expectation**and**independent increment**. - The concept of linear combinations is applied across various mathematical disciplines, such as geometry, coding theory, and linear algebra, thus bearing considerable practical significance.