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3.5.2 Linear Combinations of Random Variables II

3.5.2 Linear Combinations of Random Variables - II

Key Concepts

  • Linear Combination: The process of forming a new random variable by summing or combining others with each multiplied by a coefficient.
  • Normal Distribution: Emphasizes that linear combinations of normally distributed variables remain normally distributed.

Detailed Explanation

Single Random Variable Combination

  • Basic Formulation: For a random variable X normally distributed with mean μ and variance σ^2, a new variable Y = aX + b is also normally distributed.
  • Expectation and Variance:
    • E(Y) = aμ + b
    • Var(Y) = a^2σ^2

Combining Two Independent Variables

  • Variables: X1 and X2 with means μ1, μ2 and variances σ1^2, σ2^2 respectively.
  • Resultant Distribution:
    • If Y = X1 + X2, then E(Y) = μ1 + μ2 and Var(Y) = σ1^2 + σ2^2.

General Case for Multiple Variables

  • Variables: X1, X2, ..., Xn with coefficients a1, a2, ..., an.
  • Linear Combination: Y = a1X1 + a2X2 + ... + anXn + b.
  • Expectation and Variance:
    • E(Y) = a1μ1 + a2μ2 + ... + anμn + b
    • Var(Y) = a1^2σ1^2 + a2^2σ2^2 + ... + an^2σn^2

Special Cases:

  • If coefficients are equal and b = 0, Y is the sum of Xi's.
  • If averaging, Y becomes the average with reduced variance.

Practical Application

  • Example Scenario:
    • Machining Tolerances: Consider two machine parts with dimensions normally distributed. The gap between them can be modeled as a linear combination of their dimensions, which is also normally distributed. This can be used to assess fit probabilities and optimal functioning conditions.
    • Finance and Investment: In financial modeling, combining asset returns to evaluate overall investment performance or risk assessment.
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