# What is the normal probability distribution function? State its properties.

The normal or Gaussian Probability Distribution is most popular and important because of its
unique mathematical properties which facilitate its application to practically any physical problem in
the real world; if not for the data’s distribution directly, then in terms of the sampling distribution,
this will be the discussion in Section 7.3. It constitutes the basis for the development of many of the
statistical methods that we will learn in the following chapters. The study of the mathematical
properties of the normal probability distribution is beyond the scope of this book; however, we shall
concentrate on its usefulness in characterizing the behavior of continuous random variables that
frequently occur in daily experience.
The normal probability distribution was discovered by Abraham De Moivre in 1733 as a way of
approximating the binomial probability distribution when the number of trials in a given experiment
is very large. In 1774, Laplace studied the mathematical properties of the normal probability
distribution. Through a historical error, the discovery of the normal distribution was attributed
to Gauss who first referred to it in a paper in 1809. In the nineteenth century, many scientists noted
that measurement errors in a given experiment followed a pattern (the normal curve of errors) that
was closely approximated by this probability distribution. The normal probability distribution is
formally defined as follows:
The universally accepted notation X~Nμσ2 is read as “the continuous random variable X is normally
distributed with a population mean μ and population variance σ2. Of course in real world problems
we do not know the true population parameters, but we estimate them from the sample mean and
sample variance. However, first, we must fully understand the normal probability distribution.
The graph of the normal probability distribution is a “bell-shaped” curve, as shown in Figure 7.3. The
constants μ and σ2 are the parameters; namely, “μ” is the population true mean (or expected value) of
the subject phenomenon characterized by the continuous random variable, X, and “σ2” is the
population true variance characterized by the continuous random variable, X. Hence, “σ”
the population standard deviation characterized by the continuous random variable X; and the points
located at μ−σ and μ+σ are the points of inflection; that is, where the graph changes from cupping up
to cupping down