Linear Transformations of Random Variables (College Board AP® Statistics)

Linear transformations of random variables

What is a linear transformation of a random variable?

• A linear transformation of a random variable is where every value of the variable is either multiplied by a constant or added to another constant or a combination of both

  • This includes dividing every value by a constant

    • i.e. multiplying by a number between 0 and 1

  • This includes subtracting a constant from every value

    • i.e. adding a negative number

• If is a random variable, then a linear transformation is of the form

  • where and are real constants

• Linear transformations can be used to make the numbers more manageable

  • e.g. if takes values between 1,000 and 1,100 then the variable could be transformed by subtracting 1,000 from each value

How do linear transformations affect the mean of a random variable?

• The mean of a random variable is affected by both the multiplication of a constant and the addition of a constant

• To find the mean of the transformed variable

  • treat the original mean just like a value of the variable

    • multiply it by the constant

    • and then add the other constant

• If then

How do linear transformations affect the standard deviation of a random variable?

• The standard deviation of a random variable is only affected by the multiplication of a constant

  • It is not affected by the addition of a constant

• To find the standard deviation of the transformed variable

  • multiply the original standard deviation by the absolute value of the constant

    • this is because the standard deviation cannot be negative

• If then

How do linear transformations affect the variance of a random variable?

• The variance of a random variable is only affected by the multiplication of a constant

  • It is not affected by the addition of a constant

• To find the variance of the transformed variable

  • multiply the original variance by the square of the constant

• If then

How do linear transformations affect the shape of the distribution of a random variable?

• Addition of a constant does not change the shape of the distribution of a random variable

• Multiplication by a constant may change the shape of a distribution

  • If the random variable is multiplied by a positive constant

    • the shape remains the same

  • If the random variable is multiplied by a negative constant

    • the shape is flipped horizontally