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normal random variable∗
Koro†
2013-03-21 12:40:34
For any real numbers µ and σ > 0, the Gaussian probability distribution
function with mean µ and variance σ 2 is defined by
2 1
exp − 21 x−µ
.
f (x) = √
σ
2πσ 2
When µ = 0 and σ = 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 ∼ N (µ, σ 2 ). It has expected value
µ, and variance σ 2 .
Cumulative distribution function
The cumulative distribution function of a standard normal variable, often denoted by
Z z
2
1
e−x /2 dx ,
Φ(z) = √
2π −∞
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.
∗ hNormalRandomVariablei
created: h2013-03-21i by: hKoroi version: h30527i Privacy
setting: h1i hDefinitioni h62E15i h60E05i h05C50i h34K05i
† This text is available under the Creative Commons Attribution/Share-Alike License 3.0.
You can reuse this document or portions thereof only if you do so under terms that are
compatible with the CC-BY-SA license.
1
Figure 1: Graph of densities of the normal distribution for various values of the
standard deviation σ
Properties
µ
σ2
0
3
MX (t) = exp µt + (σt)2 /2
φX (t) = exp µit − (σt)2 /2
Mean
Variance
Skewness
Kurtosis
Moment-generating function
Characteristic function
• If Z is a standard normal random variable, then X = σZ +µ is distributed
as N (µ, σ 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.
2