Last Updated: Mar 27, 2024
Difficulty: Easy

# Normal distribution

Leveraging ChatGPT - GenAI as a Microsoft Data Expert
Speaker
Prerita Agarwal
Data Specialist @
23 Jul, 2024 @ 01:30 PM

## Introduction

The Normal Distribution is also known as the Gaussian Distribution and Bell-Shaped Distribution. When a random experiment is replicated, the random variable that will equal the average or total result over the replicates will have a normal distribution as the number of replicates becomes large.

## Probability Density Function

For âˆ’âˆž<Î¼<âˆž and Ïƒ>0, the Normal Distribution is denoted by N(Î¼,Ïƒ2), and its probability density is given by

.

Ïƒ is the Standard Deviation, and Î¼ is the Mean.

Letâ€™s break the formula into smaller pieces to understand whatâ€™s happening.

Z-score - It measures how many standard deviations away a data point lies from the mean.

.

The exp in the above probability density formula is the square of the z-score times -Â½. The values that are away from the mean have a lower probability than those near the mean. The values away from the mean have a higher z-score and thus a lower probability since the exp is negative. The vice-versa is for the values closer to the mean.

Now comes the 68-95-99.7 rule. The rule states that the % of values within a band around the mean in a normal distribution having a width of two, four, and six standard deviations comprise 68%, 95%, and 99.7% of all the values.

Source

Source

The effects of Ïƒ and Î¼ on the Distribution are shown in the above picture.

The above f(x) is integrated into one, which can be proved by the Gaussian integral below.

Source

Let â€˜Xâ€™ = normal distributed random variable with parameters Î¼ and Ïƒ^2. The area inside the normal distribution curve is 1 as the probability is 1.

Thus, = 1;

writing x as (x-Î¼ ) + Î¼  yields

Letting y=x-Î¼

The first part is symmetric about the y-axis, hence its value is 0.

Thus,

Expectation E[x] = Î¼

Variance = Ïƒ^2

Standard Deviation = .

Get the tech career you deserve, faster!
Connect with our expert counsellors to understand how to hack your way to success
User rating 4.7/5
1:1 doubt support
95% placement record
Akash Pal
Senior Software Engineer
326% Hike After Job Bootcamp
Himanshu Gusain
Programmer Analyst
32 LPA After Job Bootcamp
After Job
Bootcamp

## Standard Normal Distribution

When the mean is set to 0, and the standard deviation is set to 1 in the General Normal Distribution, the corresponding Distribution is called the Standard Normal Distribution.

The probability density function now equals:-

We do not get a closed formula by the cumulative density function of normal distribution. Thus there are pre-computed values in tables that can be used whenever needed. The values stored in this table are only for the standard Distribution. So if we want to find the cumulative probability for the general normal Distribution, we need to standardize and then compute using the value table.

Source

If X is a normal random variable with E(X)=Î¼ and V(X)=Ïƒ^2

the random variable is a normal random variable with E(Z)=0 and V(Z)=1.

That is, Z is a standard normal random variable.

The expected value of a standard normal random variable X is

Expected value

Variance

Standard Deviation = 1.

## FAQs

1. Are normal distributions symmetrical?
Normal Distributions are always symmetrical. Not all symmetrical distributions are normal.

2. Name the two common parameters of normal Distribution.
The two common parameters of normal Distribution are:- Mean and Average

3. What is standard normal Distribution?
Normal Distribution having a mean 0 and standard deviation of 1 is called Standard Normal Distribution.

4. Can Normal Distribution be discrete?
In some cases, Normal Distribution can be used to approximate discrete data.

5. How many standard normal distributions are there?
There is only one standard normal Distribution.

## Key Takeaways

In this article, we have extensively discussed Normal Distribution and Standard Normal Distribution. We hope this blog has helped you enhance your knowledge regarding Normal Distribution.

If you want to learn more, check out our articles on Poisson distributionProsecutorâ€™s fallacy.

Refer to our guided paths on Coding Ninjas Studio to learn more about DSA, Competitive Programming, JavaScript, System Design, etc.

Enrol in our courses and refer to the mock test and problems available.

Take a look at the interview experiences and interview bundle for placement preparations.

Do upvote our blog to help other ninjas grow.

Happy Coding!

Topics covered
1.
Introduction
2.
Probability Density Function
3.
Standard Normal Distribution
4.
FAQs
5.
Key Takeaways