Sebaran normal: Béda antarrépisi
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'''Normal distribution''' (distribusi normal) mangrupakeun hal anu penting dina [[probability distribution]] di loba widang.
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=== Fungsi probabiliti densiti ===
[[Fungsi dénsitas probabilitas]] dina '''sebaran normal''' numana mean
:<math>f(x) = {1 \over \sigma\sqrt{2\pi} }\,e^{-{(x-\mu )^2 / 2\sigma^2}}</math>
(Tempo oge [[exponential function|fungsi eksponensial]] jeung [[pi]].) Lamun [[variabel acak]] ''X'' ngabogaan distribusi ieu, bisa dituliskeun ''X'' ~ N(
:<math>f(x) = {1 \over \sqrt{2\pi} }\,e^{-{x^2 / 2}}</math>
Gambar diluhur nunjukeun grafik probability density function tina sebaran normal numana
For all normal distributions,
the density function is symmetric about its mean value. About 68% of the area under the curve is within one standard deviation of the mean, 95.5% within two standard deviations, and 99.7% within three standard deviations. The [[inflection point
=== Fungsi Sebaran Kumulatif ===
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The following graph shows the cumulative distribution function for values of ''z'' from -4 to +4:
[[
On this graph, we see the probability that a standard normal variable has a value less than 0.25 is approximately equal to 0.60.
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== Pasipatan ==
# Lamun ''X'' ~ N(
# If ''X''<sub>1</sub> ~ N(
# If ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> are [[Statistical independence|independent]] standard normal variables, then ''X''<sub>1</sub><sup>2</sup> + ... + ''X''<sub>''n''</sub><sup>2</sup> has a [[sebaran chi-kuadrat]] with ''n'' degrees of freedom.
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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
:<math> Z = \frac{X - \mu}{\sigma} </math>
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:<math>X=\sigma Z+\mu \,</math>
is a normal random variable with mean
The standard normal distribution has been tabulated, and the other normal distributions are simple transformations of the standard one.
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=== Generating normal random variables ===
For computer simulations, it is often useful to generate
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]].
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The practical importance of the central limit theorem is that the normal distribution can be used as an approximation to some other distributions.
* [[Sebaran binomial]] mibanda parameter ''n'' sarta ''p'' ngadeukeutan kana normal keur ''n'' nu badag sarta ''p'' teu deukeut ka 1 atawa 0. ''Pendekatan'' sebaran normal mibanda mean
* A [[Poisson distribution]] with parameter
Whether these approximations are sufficiently accurate depends on the purpose for which they are needed, and the rate of convergence to the normal distribution.
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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 family|reproductive random variables]] are involved, such as
** Binomial random variables, associated to yes/no questions;
** Poisson random variables, associates 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-
** 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 statistics|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 law]]s rather than normality.
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=== Financial variables ===
Because of the exponential nature of [[interest]] and [[inflation]],
[[Benoît Mandelbrot]], the popularizer of [[fractals]], has claimed that even the assumption of lognormality is flawed.
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== Tempo oge ==
* [[multivariate normal distribution]].
== Tumbu kaluar jeung rujukan ==
* [http://ce597n.www.ecn.purdue.edu/CE597N/1997F/students/michael.a.kropinski.1/project/tutorialMichael A. Kropinski's normal distribution tutorial]
* S. M.Stigler: ''Statistics on the Table'', Harvard University Press 1999, chapter 22. History of the term "normal distribution".
* [http://members.aol.com/jeff570/mathword.html Earliest Known uses of some of the Words of Mathematics]. See: [http://members.aol.com/jeff570/n.html] for "normal", [http://members.aol.com/jeff570/g.html] for "Gaussian", and[http://members.aol.com/jeff570/e.html] for "error".
* [http://members.aol.com/jeff570/stat.html Earliest Uses of Symbols in Probability and Statistics]. See Symbols associated with the Normal Distribution.
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[[ar:توزيع احتمالي طبيعي]]
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[[id:Distribusi normal]]
[[is:Normaldreifing]]
[[it:
[[ja:正規分布]]
[[ko:정규분포]]
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