Normal distribution: Definition, pdf, properties with applications
In probability, normal distribution is the most important continuous distribution in statistics because its common in natural phenomena. It is also known as Gaussian distribution and always symmetric about mean. There are also various probability distributions such as Bernoulli, Binomial , Negative Binomial, Geometric, Hypergeomettric, Poisson, Logarithmic series, Power series, Gamma, Beta, Uniform , Exponential distribution etc. But from all this distribution normal is the best mesure.
Normal curve( Bell shaped) 
The PDF, CDF, CF, MGF of normal distribution
The cumulative density function (cdf) is,
CDF (cumulative density function)


Mean, Median, Mode 
μ 
Varience 

Skewness, Kurtosis 
0 
MGF (moment generating function) 

CF 

Why normal distribution is so common in natural phenomena?
Properties of the Normal Distribution
 The distribution is symmetric about the point x = μ and has a characteristic bellshaped curve with respect to it. Therefore, its skewness is equal to zero i.e. the curve is neither inclined to the right (negatively skewed) nor to the left (positively skewed).
 The mean, median and the mode of a normal distribution, all coincide with each other and are equal to μ.
 The Standard Deviation for this distribution is equal to σ.
First Quartile: μ – 0.675σ and the Third Quartile: μ + 0.675σ
Thus, Quartile Deviation: 0.675
Practical Applications of the Standard Normal Model
The standard normal distribution could help you figure out which subject you are getting good grades in and which subjects you have to exert more effort into due to low scoring percentages. Once you get a score in one subject that is higher than your score in another subject, you might think that you are better in the subject where you got the higher score. This is not always true. You can only say that you are better in a particular subject if you get a score with a certain number of standard deviations above the mean. The standard deviation tells you how tightly your data is clustered around the mean; It allows you to compare different distributions that have different types of data — including different means. For example, if you get a score of 90 in Math and 95 in English, you might think that you are better in English than in Math. However, in Math, your score is 2 standard deviations above the mean. In English, it’s only one standard deviation above the mean. It tells you that in Math, your score is far higher than most of the students (your score falls into the tail). Based on this data, you actually performed better in Math than in English!..(source)Skewness and Kurtosis
The skewness and kurtosis coefficients always measure how different a given distribution is from a normal distribution. The skewness measures the symmetry of a distribution. The normal distribution is symmetric and has zero skewness . If the distribution of a data set has negative skewness (skewness<0), then the left tail of the distribution is longer than the right tail and if skewness>0 then the right tail of the distribution is longer than the left.
 kurtosis=3 then mesokurtic
 kurtosis<3 then platykurtic
 kurtosis>3 then leptokurtic
 Normal distribution is always in a bell shape that means mesokurtic.
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