Normal Distribution - Definition

Definition

The simplest case of a normal distribution is known as the standard normal distribution, described by this probability density function:

\phi(x) = \frac{1}{\sqrt{2\pi}}\, e^{- \frac{\scriptscriptstyle 1}{\scriptscriptstyle 2} x^2}.

The factor in this expression ensures that the total area under the curve ϕ(x) is equal to one, and 1/2 in the exponent makes the "width" of the curve (measured as half the distance between the inflection points) also equal to one. It is traditional in statistics to denote this function with the Greek letter ϕ (phi), whereas density functions for all other distributions are usually denoted with letters f or p. The alternative glyph φ is also used quite often, however within this article "φ" is reserved to denote characteristic functions.

Every normal distribution is the result of exponentiating a quadratic function (just as an exponential distribution results from exponentiating a linear function):

f(x) = e^{a x^2 + b x + c}. \,

This yields the classic "bell curve" shape, provided that a < 0 so that the quadratic function is concave for x close to 0. f(x) > 0 everywhere. One can adjust a to control the "width" of the bell, then adjust b to move the central peak of the bell along the x-axis, and finally one must choose c such that (which is only possible when a < 0).

Rather than using a, b, and c, it is far more common to describe a normal distribution by its mean μ = − b/2a and variance σ2 = − 1/2a. Changing to these new parameters allows one to rewrite the probability density function in a convenient standard form:

f(x) = \frac{1}{\sqrt{2\pi\sigma^2}}\, e^{\frac{-(x-\mu)^2}{2\sigma^2}} = \frac{1}{\sigma}\, \phi\!\left(\frac{x-\mu}{\sigma}\right).

For a standard normal distribution, μ = 0 and σ2 = 1. The last part of the equation above shows that any other normal distribution can be regarded as a version of the standard normal distribution that has been stretched horizontally by a factor σ and then translated rightward by a distance μ. Thus, μ specifies the position of the bell curve's central peak, and σ specifies the "width" of the bell curve.

The parameter μ is at the same time the mean, the median and the mode of the normal distribution. The parameter σ2 is called the variance; as for any random variable, it describes how concentrated the distribution is around its mean. The square root of σ2 is called the standard deviation and is the width of the density function.

The normal distribution is usually denoted by N(μ, σ2). Thus when a random variable X is distributed normally with mean μ and variance σ2, we write

X\ \sim\ \mathcal{N}(\mu,\,\sigma^2). \,

Read more about this topic:  Normal Distribution

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