# Linear combinations of any random variables

## Basic Definition

• A linear combination of random variables consists of any sum of the random variables, each multiplied by a constant.
• It applies to both discrete and continuous random variables.
• The constants are not random, and are usually chosen based on specific conditions or constraints.

## Linear Combinations and Expected Values

• If X and Y are random variables, and a and b are constants, the expected value of the linear combination is given by E(aX + bY) = aE(X) + bE(Y).
• Expected value is a type of average value, it gives a measure of the “center” of the distribution of the variable.
• This property can make computations easier, particularly in statistics and probability.

## Linear Combinations and Variances

• The variance of the linear combination is given by Var(aX + bY) = a^2 Var(X) + b^2 Var(Y) + 2abCov(X,Y) if X and Y are not independent.
• However, if X and Y are independent variables, the equation simplifies to Var(aX + bY) = a^2 Var(X) + b^2 Var(Y) because Cov(X,Y) is 0.
• Variance gives a measure of how much the values of the random variable vary around the expected value.

## Distributions of Linear Combinations

• The distribution of the linear combination of two independent random variables can be determined by their convolution.
• The convolution of the distribution functions of the two variables will provide the distribution function of their linear combination.
• In the specific case where the random variables are normally distributed, their linear combination is also normally distributed.

## Key Properties

• Linear combinations of random variables preserve the properties Linearity of Expectation and independent increment.
• Linear combinations are extensively used in numerous areas of mathematics, including geometry, coding theory, and linear algebra, endowing them with major practical importance.