Linear Combinations of Random Variables (DP IB Maths: AI HL)

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Dan

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Dan

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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
    • Var left parenthesis X right parenthesis equals straight E open parentheses X squared close parentheses minus open square brackets straight E open parentheses X close parentheses close square brackets squared
      • where straight E open parentheses X squared close parentheses equals sum x squared straight P left parenthesis X equals x right parenthesis
    • 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
    • X over a equals 1 over a X

Worked example

X is a random variable such that straight E left parenthesis X right parenthesis equals 5and Var left parenthesis X right parenthesis equals 4.

Find the value of:

(i)
straight E left parenthesis 3 X plus 5 right parenthesis
(ii)
Var left parenthesis 3 X plus 5 right parenthesis
(iii)
Var left parenthesis 2 minus X right parenthesis.

4-4-2-ib-aa-ai-hl-axb-we-solution

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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)
  • 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, ..., Xn be n random variables and a1, a2, ..., an be n constants

straight E invisible function application open parentheses a subscript 1 X subscript 1 plus-or-minus a subscript 2 X subscript 2 plus-or-minus blank horizontal ellipsis plus-or-minus a subscript n X subscript n close parentheses equals a subscript 1 straight E invisible function application open parentheses X subscript 1 close parentheses plus-or-minus a subscript 2 straight E invisible function application open parentheses X subscript 2 close parentheses plus-or-minus blank horizontal ellipsis plus-or-minus a subscript n straight E invisible function application open parentheses X subscript n close parentheses

    • This is given in the formula booklet
    • This can be written as straight E invisible function application open parentheses sum a subscript i X subscript i close parentheses equals sum a subscript i straight E invisible function application open parentheses X subscript i close parentheses blank
    • This is true for any random variable

Var invisible function application open parentheses a subscript 1 X subscript 1 plus-or-minus a subscript 2 X subscript 2 plus-or-minus blank horizontal ellipsis plus-or-minus a subscript n X subscript n close parentheses equals a subscript 1 ² Var invisible function application open parentheses X subscript 1 close parentheses plus a subscript 2 ² Var invisible function application open parentheses X subscript 2 close parentheses plus blank horizontal ellipsis plus a subscript n ² Var invisible function application open parentheses X subscript n close parentheses

    • This is given in the formula booklet
    • This can be written as Var invisible function application open parentheses sum a subscript i X subscript i close parentheses equals sum a subscript i superscript 2 Var invisible function application open parentheses X subscript i close parentheses
    • 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

Examiner Tip

  • In an exam when dealing with multiple variables ask yourself which of the two cases is true
    • You are adding together different observations using the same variable: X1X+ ... + Xn
    • You are taking a single observation of a variable and multiplying it by a constant: nX

Worked example

X and Yare independent random variables such that 

straight E left parenthesis X right parenthesis equals 5Var left parenthesis X right parenthesis equals 3,

straight E left parenthesis Y right parenthesis equals negative 2Var left parenthesis Y right parenthesis equals 4.

Find the value of:

(i)
straight E left parenthesis 2 X plus 5 Y right parenthesis,
(ii)
Var left parenthesis 2 X plus 5 Y right parenthesis,
(iii)
Var left parenthesis 4 X minus Y right parenthesis.

4-6-1-ib-ai-hl-axby-we-solution

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Dan

Author: Dan

Expertise: Maths

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.