Linear Combinations of Random Variables (DP IB Applications & Interpretation (AI): HL): Revision Note

Transformation of a single variable

What is Var(X)?

• Var(X) represents the variance of the random variable X

• Var(X) can be calculated by the formula

  • $\mathrm{Var}\left(X\right)=\mathrm{E}\left(X^2\right)-{\left[\mathrm{E}\left(X\right)\right]}^2$

    • where $\mathrm{E}\left(X^2\right)=\sum x^2\mathrm{P}(X=x)$

  • You will not be required to use this formula in the exam

What are the formulae for E(aX ± b) and Var(aX ± b)?

• If a and b are constants then the following formulae are true:

  • E(aX ± b) = aE(X) ± b

  • Var(aX ± b) = a² Var(X)

    • These are given in the exam formula booklet

• This is the same as linear transformations of data

  • The mean is affected by multiplication and addition/subtraction

  • The variance is affected by multiplication but not addition/subtraction

Transformation of multiple variables

What is the mean and variance of aX + bY?

• Let X and Y be two random variables and let a and b be two constants

• E(aX + bY) = aE(X) + bE(Y)

  • This is true for any random variables X and Y

• Var(aX + bY) = a² Var(X) + b² Var(Y)

  • This is true if X and Y are independent

• E(aX - bY) = aE(X) - bE(Y)

  • This is true for any random variables X and Y

• Var(aX - bY) = a² Var(X) + b² Var(Y)

  • This is true if X and Y are independent

  • Notice that you still add the two terms together on the right hand side

    • This is because b² is positive even if b is negative

    • Therefore the variances of aX + bY and aX - bY are the same

What is the mean and variance of a linear combination of n random variables?

• The results for two random variables can be extended to any number of random variables

• Let X₁, X₂, ..., X_n be n random variables and a₁, a₂, ..., a_n be n constants
   

  $\mathrm{E}\left(a_1{X}_1\pm a_2{X}_2\pm ...\pm a_n{X}_n\right)=a_{\mathit{1}}\mathrm{E}\mathit{\left(}{\mathit{X}}_{\mathit{1}}\mathit{\right)}\pm a_{\mathit{2}}\mathrm{E}\mathit{\left(}{X}_{\mathit{2}}\mathit{\right)}\pm ...\pm a_n\mathrm{E}\mathit{\left(}{X}_n\mathit{\right)}$
   

  • This is given in the exam formula booklet

  • This can be written as $\mathrm{E}\left(\sum a_i{X}_i\right)=\sum a_i\mathrm{E}\left(X_i\right)$

  • This is true for any random variable 

  $\mathrm{Var}\left(a_1{X}_1\pm a_2{X}_2\pm ...\pm a_n{X}_n\right)={a_{\mathit{1}}}^2\mathrm{Var}\mathit{\left(}{X}_{\mathit{1}}\mathit{\right)}+{a_{\mathit{2}}}^{\mathit{2}}\mathrm{Var}\mathit{\left(}{X}_{\mathit{2}}\mathit{\right)}+...+{a_n}^{\mathit{2}}\mathrm{Var}\mathit{\left(}{X}_n\mathit{\right)}$
   

  • This is given in the exam formula booklet

  • This can be written as $\mathrm{Var}\left(\sum a_i{X}_i\right)=\sum a_i^2\mathrm{Var}\left(X_i\right)$

  • This is true if the random variables are independent

    • Notice that the constants get squared so the terms on the right hand side will always be positive

For a given random variable X, what is the difference between 2X and X₁ + X₂?

• 2X means one observation of X is taken and then doubled

• X₁ + X₂ means two observations of X are taken and then added together

• 2X and X₁ + X₂ have the same expected values

  • E(2X) = 2E(X)

  • E(X₁ + X₂) = E(X₁) + E(X₂) = 2E(X)

• 2X and X₁ + X₂ have different variances

  • Var(2X) = 2²Var(X) = 4Var(X)

  • Var(X₁ + X₂) = Var(X₁) + Var(X₂) = 2Var(X)

• To see the distinction:

  • Suppose X could take the values 0 and 1

    • 2X could then take the values 0 and 2

    • X₁ + X₂ could then take the values 0, 1 and 2

• Questions are likely to describe the variables in context

  • For example: The mass of a carton containing 6 eggs is the mass of the carton plus the mass of the 6 individual eggs

  • This can be modelled by M = C + E₁ + E₂ + E₃ + E₄ + E₅ + E₆ where

    • C is the mass of a carton

    • E is the mass of an egg

  • It is not C + 6E because the masses of the 6 eggs could be different