Normal Distribution

Explained:

normal distribution

standard normal distribution
 
   

Due to implications of the central limit theorem, the normal distribution is arguably the most important probability distribution in probability. It is specified by two parameters: a mean μ and variance σ2. We denote the distribution N(μ,σ2). Its probability density function (PDF) is:

 [1]

which is illustrated in Exhibit 1:

N(μ,σ2) Probability Density Function
Exhibit 1

The N(μ,σ2) distribution has a symmetric "bell shaped" probability density function.

Irrespective of its mean or standard deviation, every normal distribution has skewness and kurtosis:

[2]
   
[3]

It has been proven that there is no analytic formula for the cumulative distribution function (CDF) Φ of a normal distribution. The function exists. It simply can’t be expressed in terms of other standard functions. In practice, it and its inverse Φ–1 are approximated to many decimal places using computer algorithms.

   

A linear polynomial of a normal random variable is also normal. If X ~ N(μ,σ2), then:

bX + a ~ N(bμ + a, (bσ)2)

 [4]

for any scalars a, b. This means that any N(μ,σ2) random variable X can be expressed as a linear polynomial of some N(0,1) random variable Z:

X = σZ + μ.

 [5]

We call N(0,1) the standard normal distribution.

Related Internal Links

Brownian motion A simple continuous stochastic process that is widely used in physics and finance for modeling random behavior that evolves over time.

Cauchy distribution A bell-shaped distribution that is more peaked and has fatter tails than the normal distribution.

central limit theorem A theorem that explains why the normal distribution plays such an important role in probability theory.

chi-squared distribution If you square a normal random variable, the result is a chi-squared random variable.

Cornish-Fisher expansion A formula for approximating quintiles of a random variable based only on its first few cumulants.

expected value A parameter describing the "center of gravity" of a distribution.

joint normal distribution A multivariate distribution with normal marginal distributions.

kurtosis A parameter describing the peakedness and tails of a distribution.

lognormal distribution A random variable is lognormal if its logarithm is normal.

quantile A notion from probability that can be used as a parameter.

random walk A discrete stochastic process whose increments form a white noise.

skewness A parameter that describes the lack of symmetry of a distribution.

stable Paretian distribution A non-normal stable distribution.

standard deviation A parameter describing the dispersion of a distribution.

uniform distribution A continuous probability distribution that has constant probability on a finite interval.

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Related Books

Salsburg (2001) is a wonderful history of probability and statistics. Degroot and Schervish (2002) is a standard university text. Evans, Hastings and Peacock (2000) is a handy reference with detailed information on numerous probability distributions. All three cover the normal distribution.

Lady Tasting Tea

David Salsburg
quality

 
technical  

2001

  

Probability and Statistics

Morris H. Degroot and Mark J. Schervish

quality

 
technical  

2002

  

Statistical Distributions

M. Evans, N. Hastings, B. Peacock

quality

 
technical  

2000

  

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