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2.2.1 Linear Combinations of Random Variables

Definition

Linear Combination: A linear combination of random variables is an expression formed by taking the sum of multiple random variables, each multiplied by a constant coefficient.

General Form: image

Where Y is the new random variable, image are the original random variables, and image are the coefficients.

Examples

Example 1: Investment Portfolio

  • Situation: An investor allocates funds across different assets, with each asset's return being a random variable.
  • Random Variables: image representing the returns on three different assets.
  • Linear Combination: The total return, R, on the portfolio could be expressed as image are the weights (fractions of total investment) allocated to each asset.

Example 2: Production Costs

  • Situation: A manufacturer incurs different variable costs, which are random due to fluctuating prices and quantities.
  • Random Variables: image representing costs of materials, labor, and utilities.
  • Linear Combination: Total production cost, image , where image are the quantities of each cost component used.

Properties

Expected Value:

The expected value of a linear combination of random variables is the linear combination of their expected values.
- image

Variance:

If the random variables are independent, the variance of their linear combination is the sum of their variances multiplied by the square of the coefficients.
- image

Covariance:

For non-independent variables, covariance terms also appear in the variance calculation.

Applications

  • Risk Management: Calculating the overall risk (variance) of a portfolio in finance.
  • Quality Control: Estimating the variability in product quality due to different input variations.
  • Operations Research: Modeling the total time or cost in processes involving several stochastic components.