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normal random variable (Definition)

For any real numbers $ \mu$ and $ \sigma > 0$, the Gaussian probability distribution function with mean $ \mu$ and variance $ \sigma^2$ is defined by

$\displaystyle f(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} \exp \left( - \tfrac12 \left( \tfrac{x - \mu}{\sigma} \right) ^2 \right). $
When $ \mu=0$ and $ \sigma = 1$, it is usually called standard normal distribution.

A random variable $ X$ having distribution density $ f$ is said to be a normally distributed random variable, denoted by $ X\sim N(\mu,\sigma^2)$. It has expected value $ \mu$, and variance $ \sigma^2$.

Cumulative distribution function

The cumulative distribution function of a standard normal variable, often denoted by

$\displaystyle \Phi(z) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^z e^{-x^2/2} dx\„ $
cannot be calculated in closed form in terms of the elementary functions, but its values are tabulated in most statistics books and here, and can be computed using most computer statistical packages and spreadsheets.

Uses of the Gaussian distribution

The normal distribution is probably the most frequently used distribution. Its graph looks like a bell-shaped function, which is why it is often called bell distribution.

The normal distribution is important in probability theory and statistics. Empircally, many observed distributions, such as of people's heights, test scores, experimental errors, are found to be more or less to be Gaussian. And theoretically, the normal distribution arises as a limiting distribution of averages of large numbers of samples, justified by the central limit theorem.

Figure: Graph of densities of the normal distribution for various values of the standard deviation $ \sigma$
\includegraphics{bell-curve}

Properties

Mean $ \mu$
Variance $ \sigma^2$
Skewness 0
Kurtosis 3
Moment-generating function $ M_X(t) = \exp\bigl(\mu t + (\sigma t)^2/2 \bigr) $
Characteristic function $ \phi_X(t) = \exp\bigl(\mu i t -(\sigma t)^2/2 \bigr) $
  • If $ Z$ is a standard normal random variable, then $ X = \sigma Z + \mu$ is distributed as $ N(\mu, \sigma^2)$, and conversely.
  • The sum of any finite number of independent normal variables is itself a normal random variable.

Relations to other distributions

  1. The standard normal distribution can be considered as a Student-t distribution with infinite degrees of freedom.
  2. The square of the standard normal random variable is the chi-squared random variable of degree 1. Therefore, the sum of squares of $ n$ independent standard normal random variables is the chi-squared random variable of degree $ n$.



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"normal random variable" is owned by Koro. [ full author list (5) | owner history (3) ]
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See Also: area under Gaussian curve

Other names:  normal distribution, standard normal distribution, bell distribution, bell curve, Gaussian

Attachments:
proof that normal distribution is a distribution (Proof) by Wkbj79
table of probabilities of standard normal distribution (Definition) by CWoo
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Cross-references: degree, chi-squared random variable, square, degrees of freedom, infinite, independent, finite, sum, characteristic function, kurtosis, skewness, central limit theorem, averages, scores, heights, function, graph, statistics, elementary functions, closed form, variable, density, distribution, random variable, variance, mean, probability distribution function, real numbers
There are 48 references to this entry.

This is version 16 of normal random variable, born on 2001-10-26, modified 2008-03-08.
Object id is 527, canonical name is NormalRandomVariable.
Accessed 45197 times total.

Classification:
AMS MSC: 60E05 (Probability theory and stochastic processes :: Distribution theory :: Distributions: general theory)
 62E15 (Statistics :: Distribution theory :: Exact distribution theory)
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