Normal distribution

موضوع: Normal distribution الجمعة نوفمبر 19, 2010 7:51 pm

b]Normal distribution

Probability density function[/b]

The continuous probability density function of the normal distribution is the Gaussian function

تكبير الصورةتصغير الصورة تم تعديل ابعاد هذه الصورة. انقر هنا لمعاينتها بأبعادها الأصلية.
$Normal distribution E8cf730ec1a7587ee84403dbc1c64008$
where σ > 0 is the standard deviation, the real parameter μ is the expected value, and
$Normal distribution 4a13599010e5b2223d470324aded2980$
is the density function of the "standard" normal distribution: i.e., the normal distribution with μ = 0 and σ = 1. The integral of $Normal distribution Dc9632c51d1d882c947182e4bae14b72$ over the real line is equal to one as shown in the Gaussian integral article.
As a Gaussian function with the denominator of the exponent equal to 2, the standard normal density function $Normal distribution 6270e0b9f032707ae0de870f76533e6f$ is an eigenfunction of the Fourier transform.Cumulative distribution function

The cumulative distribution function (cdf) of a probability distribution, evaluated at a number (lower-case) x, is the probability of the event that a random variable (capital) X with that distribution is less than or equal to x. The cumulative distribution function of the normal distribution is expressed in terms of the density function as follows:

تكبير الصورةتصغير الصورة تم تعديل ابعاد هذه الصورة. انقر هنا لمعاينتها بأبعادها الأصلية.
$Normal distribution 0d6d29564649c2b89454b26db0fa4e06$
The standard normal cdf is just the general cdf evaluated with μ = 0 and σ = 1:

تكبير الصورةتصغير الصورة تم تعديل ابعاد هذه الصورة. انقر هنا لمعاينتها بأبعادها الأصلية.
$Normal distribution Ee6afc55216c30d88141b7036ac08fc1$
The standard normal cdf can be expressed in terms of a special function called the error function, as
$Normal distribution 003dabb870f6a1fc0521a85000ea8090$
and the cdf itself can hence be expressed as
$Normal distribution 3537f96b6dfa850f2e6fcb765a03c28c$ Generating functions

[edit] Moment generating function

The moment generating function is defined as the expected value of exp(tX). For a normal distribution, the moment generating function is

تكبير الصورةتصغير الصورة تم تعديل ابعاد هذه الصورة. انقر هنا لمعاينتها بأبعادها الأصلية.
$Normal distribution Ec3e32bd3a987126f3c3b40e239fa768$
as can be seen by completing the square in the exponent.

[edit] Cumulant generating function

The cumulant generating function is the logarithm of the moment generating function: g(t) = μt + σ²t²/2. Since this is a quadratic polynomial in t, only the first two cumulants are nonzero.

[edit] Characteristic function

The characteristic function is defined as the expected value of exp(itX), where i is the imaginary unit. So the characteristic function is obtained by replacing t with it in the moment-generating function.
For a normal distribution, the characteristic function is

تكبير الصورةتصغير الصورة تم تعديل ابعاد هذه الصورة. انقر هنا لمعاينتها بأبعادها الأصلية.
$Normal distribution 37e8462c5cc0193558226a94aa4f2a03$ Properties

Some properties of the normal distribution:

If $Normal distribution 744888b68a64964aacef3bcd8d6777e7$ and a and b are real numbers, then $Normal distribution 701cfc64a709902a1270c47864119652$ (see expected value and variance).
If $Normal distribution Eac9f1d639c15276cf22d85ff11a7b6a$ and $Normal distribution 2e26b49dbd44bb0b4650744a65f83097$ are independent normal random variables, then:
- Their sum is normally distributed with $Normal distribution 81fcbdfdcfb20f9af35db82829da43b0$ (proof).
  Interestingly, the converse holds: if two independent random variables
  have a normally-distributed sum, then they must be normal themselves —
  this is known as Cramér's theorem.
- Their difference is normally distributed with $Normal distribution 8b22e1778fd0e4548eda237bad71aa44$ .
- If the variances of X and Y are equal, then U and V are independent of each other.
- The Kullback-Leibler divergence,
  تكبير الصورةتصغير الصورة تم تعديل ابعاد هذه الصورة. انقر هنا لمعاينتها بأبعادها الأصلية.
  $Normal distribution 3343df5d9bc94fa8ced3d0f921566c57$

If $Normal distribution 1a368150e2171639da10166086a11bd3$ and $Normal distribution 9d49cf31b33111492ae2884c760231e9$ are independent normal random variables, then:

Their product XY follows a distribution with density p given by
$Normal distribution B009e9a11285d981783c4467debe66e0$ where K₀ is a modified Bessel function of the second kind.
Their ratio follows a Cauchy distribution with $Normal distribution 1fc3faa96303a0341027aacb3a8e5c5d$ . Thus the Cauchy distribution is a special kind of ratio distribution.

If $Normal distribution E8eacc087b64e56391a3b8ddf4216d77$ are independent standard normal variables, then $Normal distribution 4b82e42397459f7359433c6673601109$ has a chi-square distribution with n degrees of freedom.

If $Normal distribution E8eacc087b64e56391a3b8ddf4216d77$ are independent standard normal variables, then the sample mean $Normal distribution F8ee4c5b552319c9e1183d3d89355316$ and sample variance $Normal distribution B0a33006cefd0b7d37794e42ec213a35$ are independent. This property characterizes normal distributions (and helps to explain why the F-test is non-robust with respect to non-normality!)

[edit] 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 ~ N(μ,σ²), then
$Normal distribution F7aa6c4c6f1dc137bb57a719ca20edb0$
is a standard normal random variable: Z ~ N(0,1). An important consequence is that the cdf of a general normal distribution is therefore

تكبير الصورةتصغير الصورة تم تعديل ابعاد هذه الصورة. انقر هنا لمعاينتها بأبعادها الأصلية.
$Normal distribution 7b0e968a9422be89bc5054772359a82a$
Conversely, if Z is a standard normal distribution, Z ~ N(0,1), then
X = σZ + μ
is a normal random variable with mean μ and variance σ².
The standard normal distribution has been tabulated (usually in the
form of value of the cumulative distribution function Φ), and the other
normal distributions are the simple transformations, as described
above, 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.Moments

The first few moments of the normal distribution are:
Number Raw moment Central moment Cumulant

0	1	1
1	μ	0	μ
2	μ² + σ²	σ²	σ²
3	μ³ + 3μσ²	0	0
4	μ⁴ + 6μ²σ² + 3σ⁴	3σ⁴	0
5	μ⁵ + 10μ³σ² + 15μσ⁴	0	0
6	μ⁶ + 15μ⁴σ² + 45μ²σ⁴ + 15σ⁶	15σ⁶	0
7	μ⁷ + 21μ⁵σ² + 105μ³σ⁴ + 105μσ⁶	0	0
8	μ⁸ + 28μ⁶σ² + 210μ⁴σ⁴ + 420μ²σ⁶ + 105σ⁸	105σ⁸	0

All cumulants of the normal distribution beyond the second are zero.
Higher central moments (of order 2k with μ = 0) are given by the formula
$Normal distribution 9ee2fb62550523bac223697b7ad104b0$

موضوع: رد: Normal distribution الجمعة نوفمبر 19, 2010 8:24 pm

بارك الله فيك