Linear combinations of any random variables
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.