Linear Combinations Of Random Variables

A linear combination is any expression like aX + bY + c. In probability, these matter because totals, averages, spreads, and portfolio-style aggregates are all linear combinations of simpler random variables.

The good news is that means and variances of linear combinations follow organized rules. The bad news is that many exam mistakes happen when candidates remember the mean rule but forget what dependence does to the variance rule.

Suppose X and Y have means 4 and 6, variances 9 and 16, and covariance 3. Let Z = X + 2Y. Then E[Z] = 4 + 2(6) = 16.

For the variance, Var(Z) = 9 + 4(16) + 4(3) = 85. The key step is the covariance term. If you drop it, you get the wrong answer for exactly the reason the syllabus tests this topic: aggregation depends on dependence.

The official multivariate outcomes explicitly mention probabilities and moments for linear combinations of independent random variables, along with continuous normal random variables and CLT approximations.

That makes linear combinations a hinge topic. They connect joint distributions to covariance, covariance to normal models, and normal models to CLT-style approximations.

The main mistake is forgetting the covariance term. The second is using the variance where the standard deviation is required for later standardization. The third is assuming independence because it would make the arithmetic nicer.

Linear combinations are everywhere in statistics: weighted sums, estimators, contrasts, portfolio returns, regression predictions, and aggregate-loss approximations. Exam P covers the first serious version of that language.