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CIE A-Level Maths Study Notes

5.2.2 Variance of Linear Combinations

Variance is a key concept in statistics, especially when dealing with random variables. Students need to understand how variance is calculated in linear combinations of random variables, as it forms the basis for more complex statistical analyses.

Understanding Variance

  • Variance, denoted as Var(X)Var(X), measures the spread of a set of numbers from their mean.
  • It's the average of the squared differences from the mean.

Variance of a Single Random Variable

  • Formula: Var(aX+b)=a2Var(X)Var(aX+b) = a² Var(X), where 'a' and 'b' are constants, X is the random variable.
  • 'b' does not affect variance, only 'a' does.

Example Calculation: Var(3X + 2)

Consider a random variable XX with a known variance, say Var(X)=4Var(X) = 4. If we transform XX using a linear equation, for example, 3X+23X + 2, the variance of this new variable is calculated as follows:

  • Variance calculation: Var(3X+2)=324=36Var(3X + 2) = 3² * 4 = 36.
Variance of a Single Random Variable Graph

Variance of Linear Combinations of Two Independent Random Variables

  • Formula: Var(aX+bY)Var(aX+bY) = a2Var(X)+b2Var(Y)a² Var(X) + b² Var(Y), where 'a' and 'b' are constants, X and Y are independent random variables.

Example Calculation: Var(2X + 4Y)

Suppose we have two independent random variables, XX with Var(X)=5Var(X) = 5, and YY with Var(Y)=3Var(Y) = 3. For a linear combination, say 2X + 4Y, the variance is calculated as follows:

  • Variance calculation: Var(2X+4Y)=22(5)+42(3)=68Var(2X + 4Y) = 2²(5) + 4²(3) = 68.
Variance of Linear Combinations of Two Independent Random Variables Graph
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
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Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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