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Home > Normal distribution

3.1 Standardizing normal random variables

As a consequence of Property 1, it is possible to relate all normal random variables to the standard normal.

If X is a normal random variable with mean μ and variance σ², then

is a standard normal random variable: Z~N(0,1). An important consequence is that the cdf of a general normal distribution is therefore

Conversely, if Z is a standard normal random variable, then

is a normal random variable with mean μ and variance σ².

The standard normal distribution has been tabulated, and the other normal distributions are simple transformations of the standard one. Therefore, one can use tabulated values of the cdf of the standard normal distribution to find values of the cdf of a general normal distribution.

3.2 Generating normal random variables

For computer simulations, it is often useful to generate values that have a normal distribution. There are several methods; the most basic is to invert the standard normal cdf. More efficient methods are also known. One such method is the Box-Muller transform. The Box-Muller transform takes two uniformly distributed values as input and maps them to two normally distributed values. This requires generating values from a uniform distribution, for which many methods are known. See also random number generators.

The Box-Muller transform is a consequence of Property 3 and the fact that the chi-square distribution with two degrees of freedom is an exponential random variable (which is easy to generate).

3.3 The central limit theorem

The normal distribution has the very important property that under certain conditions, the distribution of a sum of a large number of independent variables is approximately normal. This is the so-called central limit theorem.

The practical importance of the central limit theorem is that the normal distribution can be used as an approximation to some other distributions.

• A binomial distribution with parameters n and p is approximately normal for large n and p not too close to 1 or 0 (some books recommend using this approximation only if np and n(1 − p) are both at least 5; in this case, a continuity correction should be applied). The approximating normal distribution has mean μ = np and standard deviation σ = (n p (1 - p))^1/2.

• A Poisson distribution with parameter λ is approximately normal for large λ. The approximating normal distribution has mean μ = λ and standard deviation σ = √λ.

Whether these approximations are sufficiently accurate depends on the purpose for which they are needed, and the rate of convergence to the normal distribution. It is typically the case that such approximations are less accurate in the tails of the distribution.

3.4 Infinite divisibility

The normal distributions are infinitely divisible probability distributions.

4 Occurrence

Approximately normal distributions occur in many situations, as a result of the central limit theorem. When there is reason to suspect the presence of a large number of small effects acting additively and independently, it is reasonable to assume that observations will be normal. There are statistical methods to empirically test that assumption.

Effects can also act as multiplicative (rather than additive) modifications. In that case, the assumption of normality is not justified, and it is the logarithm of the variable of interest that is normally distributed. The distribution of the directly observed variable is then called log-normal.

Finally, if there is a single external influence which has a large effect on the variable under consideration, the assumption of normality is not justified either. This is true even if, when the external variable is held constant, the resulting marginal distributions are indeed normal. The full distribution will be a superposition of normal variables, which is not in general normal. This is related to the theory of errors (see below).

To summarize, here's a list of situations where approximate normality is sometimes assumed. For a fuller discussion, see below.

• In counting problems (so the central limit theorem includes a discrete-to-continuum approximation) where reproductive random variables are involved, such as
  • Binomial random variables, associated to yes/no questions;
  • Poisson random variables, associated to rare events ;
• In physiological measurements of biological specimens:
  • The logarithm of measures of size of living tissue (length, height, skin area, weight);
  • The length of inert appendages (hair, claws, nails, teeth) of biological specimens, in the direction of growth; presumably the thickness of tree bark also falls under this category;
  • Other physiological measures may be normally distributed, but there is no reason to expect that a priori;
• Measurement errors are assumed to be normally distributed, and any deviation from normality must be explained;
• Financial variables
  • The logarithm of interest rates, exchange rates, and inflation; these variables behave like compound interest, not like simple interest, and so are multiplicative;
  • Stock-market indices are supposed to be multiplicative too, but some researchers claim that they are log-Lévy variables instead of lognormal;
  • Other financial variables may be normally distributed, but there is no reason to expect that a priori;
• Light intensity
  • The intensity of laser light is normally distributed;
  • Thermal light has a Bose-Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.

Of relevance to biology and economics is the fact that complex systems tend to display power laws rather than normality.

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Topics: Normal Distribution Function Standard Random Variable Variables...

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