What is the difference between a normal profile of a

Sinead Mcgee 2021-10-20 Answered
What is the difference between a normal profile of a random variable and normal pdf of a random variable. What is the median value of a random variable having a normal pdf?

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Expert Answer

Isma Jimenez
Answered 2021-10-21 Author has 7605 answers
Step 1
Normal distribution is called bell curve. The bell shape indicates that values closer to the mean are more likely, and it becomes increasingly unlikely to take values far from the mean in either direction.
Step 2
We use a mathematical model with a smooth bell-shaped curve to describe these bell-shaped data distributions. These models are called normal curves or normal distributions. This is normal profile of a random variable.
The normal distribution is by far the most important probability distribution. One of the main reasons for that is the Central Limit Theorem (CLT). The CLT states that if you add a large number of random variables, the distribution of the sum will be approximately normal under certain conditions. The importance of this result comes from the fact that many random variables in real life can be expressed as the sum of a large number of random variables and, by the CLT, we can argue that distribution of the sum should be normal. The CLT is one of the most important results in probability and we will discuss it later on. Here, we will introduce normal random variables.
We first define the standard normal random variable. We will then see that we can obtain other normal random variables by scaling and shifting a standard normal random variable.
A continuous random variable Z is said to be a standard normal (standard Gaussian) random variable, shown as \(\displaystyle{Z}∼{N}{\left({0},{1}\right)}\) if its PDF is given by
\(\displaystyle{f}{Z}{\left({z}\right)}=\frac{{1}}{\sqrt{{{2}\pi}}}\sqrt{{{\exp}}}{\left\lbrace−{z}{2}\right\rbrace}\),for all \(\displaystyle{z}\in{R}\)
For normal distribution as it is symmetric distribution. Hence \(\displaystyle{m}{e}{a}{n}={m}{e}{d}{i}{a}{n}=\text{mod}{e}=\mu\).
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