Some authors[5] instead of σ2 use its reciprocal τ = σ−2, which is called the precision. This parameterization has an advantage in numerical applications where σ2 is very close to zero and is more convenient to work with in analysis as
τ is a natural parameter of the normal distribution. Another advantage of using this parameterization is in the study of
conditional distributions in multivariate normal case.
The question which normal distribution should be called the “standard” one is also answered differently by various authors. Starting from the works of Gauss the standard normal was considered to be the one with variance σ2 = 1/2:
Stigler (1982) goes even further and suggests the standard normal with variance σ2 = 1/(2π):
According to the author, this formulation is advantageous because of a much simpler and easier-to-remember formula, the fact that the pdf has unit height at zero, and simple approximate formulas for the quantiles of the distribution.
Characterization
In the previous section the normal distribution was defined by specifying its probability density function. However there are other ways to characterize a probability distribution. They include: the cumulative distribution function, the moments, the cumulants, the characteristic function, the moment-generating function, etc.
Probability density function
The probability density function (pdf) of a random variable describes the relative frequencies of different values for that random variable. The pdf of the normal distribution is given by the formula explained in detail in the previous section:
This is a proper function only when the variance σ2 is not equal to zero. In that case this is a continuous smooth function, defined on the entire real line, and which is called the “Gaussian function”.
When σ2 = 0, the density function doesn’t exist. However we can consider a generalized function that would behave
in a manner similar to the regular density function (in the sense that it defines a measure on the real line, and it can be plugged in into an integral in order to calculate expected values of different quantities):
This is the Dirac delta function, it is equal to infinity at x = μ and is zero elsewhere.
Properties:
• Function ƒ(x) is symmetric around the point x = μ, which is at the same time the mode, the median and the mean of the distribution.
• The inflection points of the curve occur one standard deviation away from the mean (i.e., at x = μ − σ and x = μ +
σ).
• The standard normal density ϕ(x) is an eigenfunction of the Fourier transform. • The function is supersmooth of order 2, implying that it is infinitely differentiable.
• The first derivative of ϕ(x) is ϕ′(x) = −x·ϕ(x); the second derivative is ϕ′′(x) = (x2 − 1)ϕ(x). More generally, the
n-th derivative is given by ϕ(n)(x) = (−1)nHn(x)ϕ(x), where Hn is the Hermite polynomial of order n.[6]
Cumulative distribution function
The cumulative distribution function (cdf) describes probabilities for a random variable to fall in the intervals of the form (−∞, x]. The cdf of the standard normal distribution is denoted with the capital Greek letter Φ (phi), and can be computed as an integral of the probability density function:
This integral can only be expressed in terms of a special function erf, called the error function. The numerical methods for calculation of the standard normal cdf are discussed below. For a generic normal random variable with mean μ and variance σ2 > 0 the cdf will be equal to
For a normal distribution with zero variance, the cdf is the Heaviside step function:
The complement of the standard normal cdf, Q(x) = 1 − Φ(x), is referred to as the Q-function, especially in engineering texts.[7][8] This represents the tail probability of the Gaussian distribution, that is the probability that a
standard normal random variable X is greater than the number x. Other definitions of the Q-function, all of which are simple transformations of Φ, are also used occasionally.[9]
Properties:
• The standard normal cdf is 2-fold rotationally symmetric around point (0, ½): Φ(−x) = 1 − Φ(x). • The derivative of Φ(x) is equal to the standard normal pdf ϕ(x): Φ′(x) = ϕ(x).
• The antiderivative of Φ(x) is: ∫ Φ(x) dx = x Φ(x) + ϕ(x).
Quantile function
The inverse of the standard normal cdf, called the quantile function or probit function, is expressed in terms of the inverse error function:
Quantiles of the standard normal distribution are commonly denoted as zp. The quantile zp represents such a value
that a standard normal random variable X has the probability of exactly p to fall inside the (−∞, zp] interval. The quantiles are used in hypothesis testing, construction of confidence intervals and Q-Q plots. The most “famous” normal quantile is 1.96 = z0.975. A standard normal random variable is greater than 1.96 in absolute value in only 5% of cases.
Characteristic function and moment generating function
The characteristic function φX(t) of a random variable X is defined as the expected value of eitX, where i is the imaginary unit, and t ∈ R is the argument of the characteristic function. Thus the characteristic function is the Fourier transform of the density ϕ(x). For a normally distributed X with mean μ and variance σ2, the characteristic function is
[10]
The moment generating function is defined as the expected value of etX. For a normal distribution, the moment generating function exists and is equal to
The cumulant generating function is the logarithm of the moment generating function:
Since this is a quadratic polynomial in t, only the first two cumulants are nonzero.
Moments
The normal distribution has moments of all orders. That is, for a normally distributed X with mean μ and variance σ
2, the expectation E|X|p exists and is finite for all p such that Re[p] > −1. Usually we are interested only in moments of integer orders: p = 1, 2, 3, ….
• Central moments are the moments of X around its mean μ. Thus, a central moment of order p is the expected value of (X − μ) p. Using standardization of normal random variables, this expectation will be equal to σ p · E[Zp], where Z is standard normal.
Here n!! denotes the double factorial, that is the product of every other number from n to 1.
• Central absolute moments are the moments of |X − μ|. They coincide with regular moments for all even orders, but are nonzero for all odd p’s.
• Raw moments and raw absolute moments are the moments of X and |X| respectively. The formulas for these moments are much more complicated, and are given in terms of confluent hypergeometric functions 1F1 and U.
These expressions remain valid even if p is not integer. See also generalized Hermite polynomials.
Order Raw moment Central moment Cumulant 1 μ 0 μ 2 μ 2 + σ2 σ 2 σ 2 3 μ 3 + 3μσ2 0 0 4 μ 4 + 6μ2σ2 + 3σ4 3σ 4 0 5 μ 5 + 10μ3σ2 + 15μσ4 0 0 6 μ 6 + 15μ4σ2 + 45μ2σ4 + 15σ6 15σ 6 0 7 μ 7 + 21μ5σ2 + 105μ3σ4 + 105μσ6 0 0 8 μ 8 + 28μ6σ2 + 210μ4σ4 + 420μ2σ6 + 105σ8 105σ 8 0