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Normal distribution - Specification of the normal distribution
There are various ways to specify a random variable. The most visual is the probability density function (plot at the top), which represents how likely each value of the random variable is. The cumulative distribution function is a conceptually cleaner way to specify the same information, but to the untrained eye its plot is much less informative (see below). Equivalent ways to specify the normal distribution are: the moments, the cumulants, the characteristic function, the moment-generating function, and the cumulant-generating function. Some of these are very useful for theoretical work, but not intuitive. See probability distribution for a discussion.
All of the cumulants of the normal distribution are zero, except the first two.
Normal distribution - Probability density function
The probability density function of the normal distribution with mean μ and variance σ2 (equivalently, standard deviation σ) is an example of a Gaussian function,
(See also exponential function and pi.)
If a random variable X has this distribution, we write X ~ N(μ,σ2). If μ = 0 and σ = 1, the distribution is called the standard normal distribution and the probability density function reduces to
The image to the right gives the graph of the probability density function of the normal distribution various parameter values.
Some notable qualities of the normal distribution:
- The density function is symmetric about its mean value.
- The mean is also its mode and median.
- 68.268949% of the area under the curve is within one standard deviation of the mean.
- 95.449974% of the area is within two standard deviations.
- 99.730020% of the area is within three standard deviations.
- 99.993666% of the area is within four standard deviations.
- The inflection points of the curve occur at one standard deviation away from the mean.
Normal distribution - Cumulative distribution function
The cumulative distribution function (cdf) is defined as the probability that a variable X has a value less than or equal to x, and it is expressed in terms of the density function as
The standard normal cdf, conventionally denoted Φ, is just the general cdf evaluated with μ = 0 and σ = 1,
The standard normal cdf can be expressed in terms of a special function called the error function, as
The inverse cumulative distribution function, or quantile function, can be expressed in terms of the inverse error function:
This quantile function is sometimes called the probit function. There is no elementary primitive for the probit function. This is not to say merely that none is known, but rather that the non-existence of such a function has been proved.
Values of Φ(x) may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, or asymptotic series.
Normal distribution - Generating functions
The moment generating function is defined as the expected value of exp(tX). For a normal distribution, it can be shown that the moment generating function is
as can be seen by completing the square in the exponent.
The characteristic function is defined as the expected value of exp(itX), where i is the imaginary unit. For a normal distribution, the characteristic function is
The characteristic function is obtained by replacing t with it in the moment-generating function.
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 Adapted from the Wikipedia article "Specification of the normal distribution", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
