The normal distribution, also called the Gaussian distribution, is a very common continuous probability distribution. It is often used to represent real-valued random variables whose actual distributions are unknown. The normal distribution depends on two parameters: the mean (μ) and the variance (σ2). It is symmetric and bell-shaped, with the mean, median and mode all being equal and located at the center. Some key properties include that approximately 68%, 95% and 99.7% of the data lies within 1, 2 and 3 standard deviations of the mean, respectively. The normal distribution was discovered independently by de Moivre and Laplace and is also associated with Gauss.

2.  The normal (or Gaussian) distribution is a very
common continuous probability distribution.
Normal distributions are important in statistics and
are often used in the natural and social sciences to
represent real-valued random variables whose
distributions are not known.
 The normal distribution is sometimes informally
called the bell curve. However, many other
distributions are bell-shaped.

3.  The terms Gaussian function and Gaussian bell
curve are also ambiguous because they sometimes
refer to multiples of the normal distribution that
cannot be directly interpreted in terms of
probabilities.

4.  The probability density of the normal distribution is:
 Here, µ is the mean or expectation of the distribution
(and also its median and mode). The parameter σ
is its standard deviation; its variance is then σ 2. A
random variable with a Gaussian distribution is said to
be normally distributed and is called a normal
deviate.

5.  If µ= 0 and σ= 1 , the distribution is called
the standard normal distribution or the unit
normal distribution denoted by N(0,1) and a
random variable with that distribution is a standard
normal deviate.

6. The Normal Probability
distribution, which is considered
the cornerstone of the modern
statistical theory, was discovered
by Abraham de Moivre as the
limiting form of the binomial
distribution by increasing n, the
number of trials, to a very large
number for a fixed value of P.
The name of Pierre S. Laplace is
also associated with the derivation
of the normal distribution.

7. The normal distribution is also
called the Gaussian distribution in
the honour of the great German
mathematician Carl F. Gauss , who
also derived its equation
mathematically as the probability
distribution of the errors
measurements.
He was karl Pearson who in 1893
called it the normal distribution
and is the best known by this
name today.

8.  A normal probability distribution depends on the
values of the parameters µ, and σ 2 the various possible
values for these two parameters will result in an
unlimited number of different normal distribution.

9.  The Z= (X - μ) / σ has zero mean and unit variance.
Every normally distributed X with mean= µ and
variance= σ 2 is therefore conventionally transformed
into a normal Z with zero mean and unit variance by
using the following expression
Z= (X - μ) / σ

10.  The main features of the normal distribution are given
below:
 A normal distribution is bell-shaped (symmetric).

11. The mean, median, and mode are equal and are
located at the center of the distribution.
The function f(x) defining the normal distribution is a
proper p.d.f. (probability density function), i.e. f(x)≥0
and the total area under the normal curve is unity.

12. The mean and variance of the normal distribution are
µ and σ 2 respectively.

13. The mean deviation of the normal distribution is
approximately 4/5 of its standard deviation.
The normal curve has points of inflection which are
equidistant from the mean.

14.  For the normal distribution, the odd order moments
about the mean are all zero and the even order
moments about the mean are given by
µ2n = (2n-1) (2n-3)… 5.3.1 σ2n
The sum of the independent normal variables is a
normal variable.
If X is N (µ, σ 2) and if Y=a+bX, then Y is N (a+bµ, b2
σ2).

15. No matter what the values of µ and σ are, areas under
normal curve remain in certain fixed proportions with
the specified number of standard deviations on either
side of µ. For example, the interval
I. Approximately 68% of the area under the curve is
between µ-σ and µ+σ.
II. Approximately 95% of the area under the curve is
between µ-2σ and µ+2σ.
III. Approximately 99.7% of the area under the curve is
between µ-3σ and µ+3σ.

16. The normal curve approaches, but never really touches
, the horizontal axis on either side of the mean towards
plus and minus infinity, that is the curve is asymptotic
to the horizontal axis as x ±∞.

17.  Chaudhary, M. S. and Kamal, S., Introduction Statistical Theory, part 1, Ilmi Kitab Khana
Urdu Bazar Lahore. (2002)
 http://highered.mheducation.com/sites/dl/free/0073521485/940175/doane4e_sample_ch
07.pdf
 http://www.amsi.org.au/ESA_Senior_Years/PDF/ContProbDist4e.pdf
 http://trojan.troy.edu/studentsupportservices/assets/documents/presentations/math_sc
ience/Math_2200_Sections_6_2-6_3.pptx
 http://www.cliffsnotes.com/math/statistics/sampling/properties-of-the-normal-curve
 http://www.mathnstuff.com/math/spoken/here/2class/90/normal.htm
 http://s3.amazonaws.com/ppt-download/normaldistribution-110307043247-
phpapp01.pptx?response-content-
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