Standard Normal Distribution - rFronteddu/general_wiki GitHub Wiki

A transformation is a way to alter every element of a distribution to get a new distribution with similar characteristics.

For example, given f(x) as a bell curve

• f(x+3) moves the graph toward the right
• f(x-3) moves the graph toward the left
• f(x * c) reduce wideness of the bell
• f(x / c) increase wideness of the bell c>1
• Standardization is a transformation that makes a distribution with E(X)=0 and Var(X) = 1
• The distribution we get after standardizing the normal distribution is called Standard Normal Distribution

One of the advantages of the Standardized Normal Distribution is that there is a CDF (called z-score table) for its most common values.

To standardize:

• Move the graph to the left or to the right until the mean is 0 (i.e. subtract μ from every element)
• Make σ = 1 by dividing each value by σ

If we denote the Standard Normal Distribution with Z: {ZN(0,1) {∀YZ (μ,σ)

• We have Z = (Y - μ) / σ, Z is the transformation to standardize