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Most random variables are found to follow the probability distribution function

All this means is that most things which can be measured quantitatively, like a population's height, the accuracy of a machine, effectiveness of a drug on fighting bacteria, etc. will occur with a probability that can be calculated according to this equation. Since most things follow this equation, this equation is considered to be the "normal" probability density. "Normal" events follow a "normal" probability distribution.

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Q: Why normal distribution is called normal distribution?

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It is called a normal distribution.

A normal data set is a set of observations from a Gaussian distribution, which is also called the Normal distribution.

It has no special name - other than a normal (or Gaussian) distribution graph.

It is called a standard normal distribution.

One.

The standard normal distribution is a normal distribution with mean 0 and variance 1.

Suppose you could call it the Gaussian Distribution or the Laplace-Gauss (not to be confused with the Laplace distribution which takes an absolute difference from the mean rather than a squared error)... however the Brits had no one to name this distribution after (not the German and French names) and because it is the ubiquitous distribution they just called it... well the NORMAL!!

le standard normal distribution is a normal distribution who has mean 0 and variance 1

The standard normal distribution is a special case of the normal distribution. The standard normal has mean 0 and variance 1.

Because very many variables tend to have the Gaussian distribution. Furthermore, even if the underlying distribution is non-Gaussian, the distribution of the means of repeated samples will be Gaussian. As a result, the Gaussian distributions are also referred to as Normal.

When its probability distribution the standard normal distribution.

No, the normal distribution is strictly unimodal.

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