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:

    • 

  • Some values of T may be equal

    • In which case you can add their probabilities together

• The mean is calculated in the same way

  • 

• 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

  • i.e. if  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

    • and then use the formulae above

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

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

  • 

  • 

    • 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 by addition/subtraction