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z Distribution
The z distribution is rigorously called the standard normal distribution in probability and statistics textbooks and in the literature. A random variable with a standard normal distribution is called a standard normal random variable. Because Z is usually used to represent a standard normal random variable, the standard normal distribution is sometimes called the z distribution for convenience by practitioners. In the following, the convention of representing a random variable with a capital letter and a specific value of the random variable with the corresponding lowercase letter is used.
The standard normal distribution is one of the most important probability distributions in probability theory and statistics. Many random phenomena follow a normal distribution or approximately a normal distribution and can be represented by a normal random variable. A normal random variable can be transformed into a standard normal random variable through standardization. Many other random variables, such as the χ2 random variables, the t random variables, and the F random variables, can be written as functions of the standard normal random variable.
Notation
The standard normal distribution is a special case of the normal distribution family. A general normal random variable, denoted by X, with mean or expected value μX and variance σ2X is denoted by X∼N(μX,σ2X). The mean μX and the variance σ2X are the parameters of the normal distribution. The positive square root σX of the variance σ2X is the standard deviation of the random variable (i.e.,
Density Function and the Standard Normal Curve
The density function of a general normal distribution or of a general normal random variable X with mean μX and variance σ2X is given by
With μZ = 0 and σ2Z = 1, the standard normal distribution or a standard normal random variable Z has a density function
A plot of the standard normal density function with the horizontal axis representing the values of z and the vertical axis representing the values of f(z) is usually called the standard normal curve. Such a plot is shown in Figure 1. The standard normal curve has its peak at z = μZ = 0. Although z might take on any real value, the standard normal curve asymptotically touches the horizontal axis at −3 and + 3. Due to its appearance, the normal curve is also usually called the bell curve or the bell-shaped curve.
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