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The simplest case of a normal distribution is known as the standard normal distribution or unit normal distribution. This is a special case when μ = 0 {\textstyle \mu =0} and σ 2 = 1 {\textstyle \sigma ^{2}=1} , and it is described by this probability density function (or density): φ ( z ) = e − z 2 2 2 π . {\displaystyle \varphi (z ...
It is possible to have variables X and Y which are individually normally distributed, but have a more complicated joint distribution. In that instance, X + Y may of course have a complicated, non-normal distribution. In some cases, this situation can be treated using copulas.
Students of statistics and probability theory sometimes develop misconceptions about the normal distribution, ideas that may seem plausible but are mathematically untrue. For example, it is sometimes mistakenly thought that two linearly uncorrelated , normally distributed random variables must be statistically independent .
The Cauchy distribution, an example of a distribution which does not have an expected value or a variance. In physics it is usually called a Lorentzian profile, and is associated with many processes, including resonance energy distribution, impact and natural spectral line broadening and quadratic stark line broadening.
An example of this is to show that the sample mean and sample variance of a normal distribution are independent statistics, which is done in the Example section below. This property (independence of sample mean and sample variance) characterizes normal distributions.
Diagram showing the cumulative distribution function for the normal distribution with mean (μ) 0 and variance (σ 2) 1. These numerical values "68%, 95%, 99.7%" come from the cumulative distribution function of the normal distribution. The prediction interval for any standard score z corresponds numerically to (1 − (1 − Φ μ,σ 2 (z)) · 2).
The equidensity contours of a non-singular multivariate normal distribution are ellipsoids (i.e. affine transformations of hyperspheres) centered at the mean. [28] Hence the multivariate normal distribution is an example of the class of elliptical distributions.
The existence of a normal number follows from (a) immediately. The proof of the existence of computable normal numbers, based on (b), involves additional arguments. All known proofs use probabilistic arguments. Dvoretzky's theorem which states that high-dimensional convex bodies have ball-like slices is proved probabilistically. No ...