Transformation of a Single Variable (DP IB Analysis & Approaches (AA)): Revision Note

Transformation of a Single Variable

How do I calculate the expected value and variance of a transformation of X?

• Suppose X is transformed by the function f to form a new variable T = f(X)

  • This means the function f is applied to all possible values of X

• Create a new probability distribution table

  • The top row contains the values 

  • The bottom row still contains the values  which are unchanged as:

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    • Some values of T may be equal so you can add their probabilities together

• The mean is calculated in the same way

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• The variance is calculated using the same formula

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Are there any shortcuts?

• There are formulae which can be used if the transformation is linear

  •  where a and b are constants

• If the transformation is not linear then there are no shortcuts

  • You will have to first find the probability distribution of T

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

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

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    • 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

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