Statistics and Probability Dictionary
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A linear transformation is a change to a
variable characterized by one or more of the following
operations: adding a constant to the variable,
subtracting a constant from the variable,
multiplying the variable by a constant,
and/or dividing the variable by a constant.
When a linear transformation is applied to a
new random variable is created.
To illustrate, let X be a random variable, and let m
and b be constants. Each of the following
examples show how a linear transformation of X defines
a new random variable Y.
- Adding a constant: Y = X + b
- Subtracting a constant: Y = X - b
- Multiplying by a constant: Y = mX
- Dividing by a constant: Y = X/m
- Multiplying by a constant and adding a constant: Y = mX + b
- Dividing by a constant and subtracting a constant: Y = X/m - b
Note: Suppose X and Z are variables, and the
between X and Z is equal to r. If a new
variable Y is created by applying a linear transformation to X,
then the correlation between Y and Z will also equal