linear-transformations-of-drvs

E(X) & Var(X) of aX+b

What does aX+b mean?

• Let represent the number of hours you work each day

  • To work out your pay, , the company triples your hours and adds £60

    • 

  • is your expected hours per day

  • is your expected pay per day

• To calculate values like and you need the formulae below

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

• If  and  are constants then  is a linear transformation of 

• These are two important formulae:

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• If values of  are multiplied by , then

  • the mean is multiplied by 

  • the variance is multiplied by

    • as variance is the square of the spread (standard deviation)

• If an amount  is added to 

  • is added to the mean

  • the variance is unchanged

    • adding  to every value won’t make them more spread out

• Sometimes it helps to rewrite expressions into the form 

  • is the same as

Do these rules work for other functions of X?

• The expectation formula works for other functions, for example

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    • Note that does not simplify to 

  • Simplify algebra before taking expectations

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• The variance formula also works for other functions, for example

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    • Where means

  • Simplify algebra before taking variances

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