Step 1

From the given information,

A random variable is a function X with a particular domain and range that takes all possible outcomes from the function of the event or distribution. The random variable may be continuous or discrete in nature and any distribution function may or may not be followed.

Step 2

To explain these bell-shaped data distributions, we use a mathematical model with a smooth bell-shape curve.

These models are called regular curves or distributions that are normal. In several different types of common measurements, they were first named "normal" because the pattern occurred. This is a normal rv profile.

The normal distribution is by far the most relevant distribution of probabilities.

The Central Limit Theorem (CLT) that we will discuss later in the book is one of the key explanations for that. To give you an idea, the CLT states that the distribution of the sum would be roughly normal under some conditions if you add a large number of random variables. The meaning of this outcome is that many random variables can be expressed as the sum of a large number of rankings in real life.

Consider a random variable X. with expected value as E(X) and variance as \(Var(X) then \frac{x - \mu}{\sigma}\)

A normalization or normal profile of the random variable is considered. Example: It is said that Random Variable X follows the Binomial distribution.\(\displaystyle{X}\sim{B}\in{\left({n},{p}\right)}\) and we know that the binomial distribution expected value is E(X) = np and Variance, Var(X) = npq. In the form, then, random variable \(X \frac{X-np}{npq}\) has a normal profile.

A random variable X is said to follow Normal distribution then X having probability distribution function

\(\displaystyle{f{{\left({x}\right)}}}=\frac{{1}}{{\sqrt{{{2}\pi\sigma}}}}{e}^{{-\frac{{1}}{{2}}{\left(\frac{{{x}-\mu}}{\sigma}\right)}^{{2}}}}{x}\in{R}\) is known as normal pdf of a random variable. where \(\displaystyle\mu\ {i}{s}\ {m}{e}{a}{n}{\quad\text{and}\quad}\sigma\) is standard deviation of the normal distribution.