Mean and Variance of Linear Combinations

For independent random variables with given means and variances, find the mean and variance of the linear combination $aX + bY$, and analyze the effect of a given covariance.

When dealing with linear combinations of independent random variables, it's crucial to understand how their means and variances combine. The mean of a linear combination of random variables is simply a linear combination of their means. Specifically, if we have two independent random variables X and Y, and constants a and b, the mean of the linear combination aX + bY is a times the mean of X plus b times the mean of Y. This is due to the linearity property of expectation in probability theory.

On the other hand, the variance of the linear combination involves not only the individual variances of the random variables but also their covariances. For independent variables, the covariance term is zero, simplifying the calculation of variance. The variance of aX + bY is a squared times the variance of X plus b squared times the variance of Y. Understanding this helps in analyzing situations where random variables are combined, making it easier to estimate the dispersion of the result, which is particularly important in fields like finance and risk management.

However, if covariance is present and is non-zero, it affects the variance of the linear combination. Analyzing how changes in covariance impact the outcome requires an appreciation of how covariance reflects the degree to which two variables vary in tandem. This understanding is foundational in statistical analysis and illustrates how relationships between variables can impact the results of statistical measures.