Var(∑AiYi): Calculating & Explaining the Formula

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SUMMARY

The formula Var(∑AiYi) = ∑(Ai^2) Var(Yi) is established under the assumption that the Yi variables are independent random variables. To demonstrate this, one can simplify the proof by first proving the case for a single variable, specifically var(AX) = A^2var(X). This proof utilizes the definition of variance in terms of first and second moments, providing a clear pathway to understanding the overall formula.

PREREQUISITES
  • Understanding of variance and its mathematical definition
  • Familiarity with independent random variables
  • Knowledge of first and second moments in probability theory
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the properties of independent random variables in probability theory
  • Learn about the definition and calculation of variance
  • Explore the concept of first and second moments in statistics
  • Review algebraic proofs related to variance and linear transformations
USEFUL FOR

Statisticians, mathematicians, and students studying probability theory who seek to deepen their understanding of variance calculations and their applications in statistical analysis.

milk
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Var( ∑AiYi)= ∑(Ai^2) Var(Yi)
Could you show why?
Thank you
 
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Your equation includes an assumption that the Y's are independent random variables. In this case, you only need to show it for one variable, i.e. var(AX)=A2var(X). This then can be shown by using the definition of var in terms of first and second moments.
 

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