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.