4.6.1 Linear Combinations of Random Variables

What is Var(X)?

• Var(X) represents the variance of the random variable X
• Var(X) can be calculated by the formula
  • 
    • where 
  • 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 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
• Remember division can be written as a multiplication
  • 

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

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

• • This is given in the formula booklet
  • This can be written as 
  • This is true for any random variable

• • This is given in the formula booklet
  • This can be written as 
  • 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