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3.5.1 Linear Combinations of Random Variables I

3.5.1 Linear Combinations of Random Variables - I

Key Concepts

  • Linear Combination: A new random variable formed by summing multiple random variables each multiplied by a constant coefficient.
  • Normal Random Variables: If the original random variables are normal and independent, the linear combination is also normally distributed.

Detailed Explanation

Basic Definitions

  • Random Variables: Let X1, X2, ..., Xn be a sequence of random variables.
  • Constants: Let a1, a2, ..., an be constants.
  • Linear Combination: A new random variable Y is defined as
    • Y = a1X1 + a2X2 + ... + anXn

Expectation and Variance

  • Expectation of Y: The expectation is linear, so

    • E(Y) = a1E(X1) + a2E(X2) + ... + anE(Xn)

  • Variance of Y:

    • If Xi's are independent,

      • Var(Y) = a1^2Var(X1) + a2^2Var(X2) + ... + an^2Var(Xn)

    • If Xi's are correlated, covariance terms are also included:

      • Var(Y) = ∑ a_i^2Var(X_i) + ∑_{i ≠ j} 2a_ia_jCov(X_i, X_j)

Normal Distribution of Y

  • Normality: If Xi's are normal, Y is also normal. This holds regardless of whether Xi's are independent or correlated.
  • Importance: This property is crucial because it simplifies the analysis of Y, allowing the use of normal distribution properties and methods.

Practical Application

  • Statistical Modeling: Understanding how linear combinations work is vital in fields like econometrics and finance, where multiple normal variables (like errors in regression models) are combined.

Example Calculation:

  • Consider X1 and X2 are normal random variables with means 10 and 20, variances 15 and 25, respectively, and they are independent.

  • Calculate E(Y) and Var(Y) for Y = 3X1 + 4X2.

    • E(Y) = 3 * 10 + 4 * 20 = 110
    • Var(Y) = 3^2 * 15 + 4^2 * 25 = 145 + 400 = 545
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