Understanding Probability Density Function and Cumulative Density Function

Probability density function and cumulative density function explained in depth with examples and use cases for statistical analysis.

### Understanding Probability Density Function (PDF) and Cumulative Density Function (CDF) in Statistics
* 📊 Probability Density Function (PDF) is a way to **smoothen histograms**, representing the **percentage of the distribution within a specific range**.
* 🔍 The y-axis in PDF represents the **percentage of the distribution in that location**, ranging from **0 to 1 (or 0% to 100%)**.
* 🔔 PDF creates a bell-shaped curve for continuous data, making it easier to **analyze and interpret distributions**.
* 📈 Cumulative Density Function (CDF) is derived from PDF, showing the **cumulative percentage of distribution up to a certain point**.
* 📉 CDF is a step function that starts at **0** and ends at **1**, with a straight line indicating a **uniform distribution**.
* 📊 CDF helps understand the distribution characteristics, such as **percentiles and cumulative probabilities**.
* 🎯 For example, a CDF value of **0.9 at 130 kg** means that **90% of the distribution is below 130 kg**.
* 💡 These functions are essential for **exploratory data analysis** and **understanding the underlying distribution of data**.

hello my name is krishna welcome to my youtube channel. today in this particular video we will be discussing about probability density function. and this is a very important concept in statistics. we'll try to understand how exactly it is created. what is the exact use of probability density function and how does it help us. now when I go to this probability density function in my previous section I have already discussed about histogram. so this is how my histogram basically looks like. I've used the feature height and weight in my data set. and this is basically my output. and I am just considering weight feature and I am trying to plot all the points in the form of histograms. if you don't know about histograms I just see my previous video in this particular playlist itself you'll be able to understand. now histogram in the y-axis actually says that how many points are basically present within this range. suppose this this is between 70 to 80 weight. okay 90 100 110 120 130. so here you can see that suppose I say between

70 to 80 weight from this particular thing. You can see it. You can say that there are five values that are present. Okay. Between 80 to 90, there are some 10 values that are present. Between 90 to 100, they are around 12 values that are present. So, somewhere like this, we can basically plot in the form of a histogram. Now, if I want to convert this into a probability density function, there will be of varies. Or you can also say that we are trying to smoothen the histogram. When we smoothen the histogram, we will be getting a curve something like this. Okay. So, when we smoothen it and this resembles like a bell curve, bell curve. Now, suppose if I have so many points based on a domain knowledge of weight. Weight is also a Gaussian distribution. So, we are actually getting this kind of bell curve, but that is not what I am trying to explain over here. We are trying to understand what exactly is a probability density function. Now, as soon as I draw this particular bell curve, this count value will get removed in the

y-axis. Okay, now this will say that percentage of the distribution. This y-axis will now become percentage of the population. Or I can in a lot say population. I'll say percentage of the distribution. Now we will try to understand what exactly it is. So I'm just going to remove all the six understand that. But if I consider this point okay, and this may be between 60 to 70. Suppose if I consider at this point, I want to find out the y-axis. It basically says that within this range, what is the total population? What is the total percentage of this distribution that are present in this location? Okay, sorry, it should not be percentage of population, but understand that how much, how what is the percentage of the distribution that are present in this particular region. Okay, percentage of distribution basically is how many point suppose here are so many points within this so many points. What is the percentage of the points that are basically located in this particular region? And suppose I found out that okay, this values point one. This basically indicates that 10% of the distribution.

is present in this location. Suppose I consider for this particular point over here. I want to find out what is the total percentage of the distribution that is present over here. Now, I will combine this region, this whole region, I will try to combine and finally I will find out, okay, 0.25 is the total percentage of the distribution that is present over here. And similarly, we do it for each and every location. Suppose this, this, this, this is what this y value basically says because there are so many confusion still students have that what is actually present in the y-axis. Understand, guys, percentage of the total distribution that are present within that particular location. Suppose if I consider at this particular point, then I will try to see this distribution part, okay, and probably it will be around 50 percent, that is 0.5. Okay, and when we combine all these things with respect to all these particular points, it will be giving us around 100 percent. This cannot go more than hundred percent, try to understand in that particular way. It cannot go that.

basically means it cannot be more than 1. It will be between 0 to 1, okay, and this is what is the PDF function. This is basically a PDF probability density function, okay. Now there is one more concept which is called as cumulative density function, which is called as CDF, right. That is little bit different when compared to PDF. Understand CDF, how it works, okay. So I am just going to rub this particular diagram. Now let us go ahead and try to understand about cumulative density function. This is my PDF case, okay. Now suppose how what is CD, our CDF as the terms is cumulative density function, okay, cumulative density function or cumulative distribution functions also you can say. Now in CDF what will happen is that suppose for this particular point that I see over here, it is point one, okay. Now suppose for this particular point I want to find out what is the total value. That basically means I have to add this point one and whatever value, suppose this is ranging towards 0.25, okay, suppose this

is ranging towards 0.25. Now, I have to add this value. When I add this value, the total will be 0.25 plus 0.1, which is nothing but 0.35. Now, my cumulative density function will be having a line like this, okay, because I'm adding these two points. And with respect to each and every point, I'll be adding, and finally, we'll be seeing a curve something like looks like this, okay. It'll be having a curve, and this is what a curve of a cumulative density function looks like. You know, at the end of the time, the slope will become a little bit straight. Why? Because the points are falling, this values will become lower. And this is how a cumulative rule function basically indicates, is basically indicating, okay. Now, you can see over here, over here, you'll be able to see 0.35, 0.55, 0.75, to 0.85, 0.9, something like that, okay. Now, understand, suppose I take 130, and I try to populate a point out over here, and

Suppose I get 90 percentage of the value is over here, with respect to this particular point. Now, what does this indicate? This indicates that 90 percentage of the distribution in this particular data set is less than 130 kgs. This is what a cumulative density function basically indicates. You know, 90 percentage of the distribution. Because why I am getting 90%? Because when I plot this particular point over here, I'm getting point lying value right. And this is all added up. You can see that it is getting added up. So, my value is basically increasing every time. This point 1, when added with 0.25, I got point 35, right? Again, when I go up, that value will also get added. I will get 0.55, 0.75, 0.8, 0.95, 0.9. Now, this point 9 basically indicates that suppose if I plot this particular point over here, I stretch it to the y-axis, it'll show us 0.9. This basically indicates that 90 percentage of the distribution is less.

than 130 kg. That also indicates that 10% of the distribution in this particular weight feature is greater than 130 kg. What a wonderful analysis, right? We are able to understand so many things with the help of cumulative density function and PDF, that is, my probability density function. Isn't it amazing, right? So, this is what exactly CDF, what exactly is PDF. There's a whole lot of use cases with respect to this. This all information is basically analysis, what we say, it has exploratory data analysis. So, yes, this was all about this particular video. I hope you like this particular video. In this particular video, we'll understand what is a probability density function and what is the cumulative density function, that is, CDF. We understood it in depth. We understood each and every thing over here. If you have any queries, please do write your queries in the comment box. I would be very happy to help you out. So, yes, this was all about this video. I hope you liked this. Please do square channels are not already subscribed. Please share with all your friends.

Whoever requires this kind of help, I'll see you in the next video. Have a great day. Thank you one and all.

Tutorial 25- Probability Density function and CDF- EDA-Data Science

I am the cofounder of iNeuron and my experience is pioneering in machine learning, deep learning, and computer vision,an educator, and a mentor, with over 10 years' experience in the industry. This is my YouTube channel where I explain various topics on machine learning, deep learning, and AI with many real-world problem scenarios. I have delivered over 30 tech talks on data science, machine learning, and AI at various meet-ups, technical institutions, and community-arranged forums. My main aim is to make everyone familiar of ML and AI.Please subscribe and support the channel. As i love new technology, all these videos are free and I promise to make more interesting content as we go ahead.
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Understanding Probability Density Function (PDF) and its Relationship with Histograms

Probability density function (PDF) is a crucial concept in statistics, created to understand the distribution of data in a smoother way than histograms.

The PDF curve is obtained by smoothing the histogram and often resembles a bell curve, especially in the case of Gaussian distributions like weight.

PDF helps in visualizing the probability of a random variable falling within a particular range, providing a more continuous representation of data compared to the discrete nature of histograms.

Understanding the Y-Axis in Statistical Distribution

The y-axis represents the percentage of the distribution within a specific location, indicating the proportion of points in that region.

It is used to determine the total percentage of the distribution in each location and should sum up to 100%.

The y-axis values cannot exceed 100%, representing the maximum percentage of the total distribution.

Understanding Probability Density Function (PDF) and Cumulative Density Function (CDF)

PDF is a probability density function that represents the likelihood of a continuous random variable, while CDF, or cumulative density function, shows the probability that the variable takes on a value less than or equal to a specific value.

In CDF, the values are accumulated progressively, creating a curve that represents the cumulative probability.

The CDF curve will eventually flatten out as the variable's values decrease, showing how the probability accumulates as the variable increases.

Understanding Cumulative Density Function (CDF) and Probability Density Function (PDF)

The speaker discusses how a CDF indicates the percentage of the distribution that is less than a certain value, using the example of 90% of the distribution being less than 130 kgs.

They also mention that the CDF helps understand that 10% of the distribution is greater than 130 kg.

The video provides an in-depth explanation of CDF and PDF, emphasizing their importance in exploratory data analysis.

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