Linear combinations of any random variables

Linear Combinations of Random Variables

A linear combination involves two or more random variables combined in such a way that the resulting variable is a summation of the original random variables multiplied by a constant.
Formally, if X and Y are two random variables and a and b are constants, then Z = aX + bY, is a linear combination of X and Y.

The expected value E(Z) of a linear combination Z = aX + bY is calculated using the formula E(Z) = aE(X) + bE(Y).
This is a direct consequence of the linearity of expectation in statistics, meaning we can add and scale expectations.
The expected value provides us with the long-run average of Z over numerous trials, giving a measure of the “average” outcome of an experiment.

The variance of a linear combination Z = aX + bY is given by Var(Z) = a^2Var(X) + b^2Var(Y) + 2abCov(X, Y).
If the random variables X and Y are independent, the covariance term Cov(X, Y) in the formula equals zero and the equation simplifies to Var(Z) = a^2Var(X) + b^2Var(Y).
The variance gives us a measure of how much the values of Z spread out from their expected value.

Linear combination of random variables is a fundamental concept in statistics and forms the mathematical basis for numerous statistical procedures including hypothesis testing, regression analysis, and factor analysis.
It helps us to link together different random variables to create new variables which can then be explored for further relationships and interdependencies.
The properties of these combinations, such as their expected values and variances, can shed light on the overall behaviour of a complex system.

Independent variables are an important idea when dealing with linear combinations of random variables. If the two variables are independent, the distribution of the sum or difference of the variables can be determined by convolving their individual distributions.
Two variables are independent if knowing the outcome of one variable does not affect the probability of outcomes for the other variable.
Independence simplifies many statistical computations and is a common assumption in many types of statistical procedures.