What is a Probability Density Function (pdf)? ("by far the best and easy to understand explanation")

[@Nothing-jo8ci]

This video is the proof that nature isn't extremely hard to understand. Some people are just better at teaching than others. Thank you very much! I'm excited to watch as much videos as I can from your channel.

[@iain_explains]

Thanks so much for your nice comment. Glad the videos are helpful!

[@anzal.j]

​@@iain_explainsFor real man... This actually helped me alot. Thanks alot

[@jayp9158]

Or better worded, school teachers simple don't care

[@vidakgrujic827]

I need to thank you, 4 years of high school + 4 years of university could not explain me in this intuitive way. I have finally understood not only PDF, I understood the idea behind integrals too. Thank you so much, I feel blessed at the moment.

[@iain_explains]

I’m so glad I could help!

[@calender3721]

It’s crazy how much better I understand once you gave the important fundamentals and graphs. I always wonder why professors skip this

[@iain_explains]

Great. I'm glad you found it helpful. I always think that a graphical understanding of mathematics is important.

[@chillyvanilly6352]

I have to say, that your explanations Sir have just blown me away! I very much appreciate this piece of knowledge that you Sir have shared with the world (especially Uni students) in such an intuitive way, especially the integral-part!! Thank you!

[@iain_explains]

Thanks for your nice comments. I'm glad you've liked the videos.

[@neerajr8182]

Simply the best explaination I've ever got on the topic. Please keep up the good work and you will be helping a lot of engineers and other people out in the world.

[@iain_explains]

Thank you. I really appreciate your comment. Analysis of randomness is fundamental to so much of modern life, but it's a challenging concept, and still quite poorly understood on the whole. Glad you liked my explanation. Did you see my video on the expected value of a random variable? kzbin.info/www/bejne/aWSXi4qqZ52VlZI

[@ella3489]

I literally wasted my 1 hour on searching this topic. You saved my time. You explained it so well. Thank you, thank you so much!

[@iain_explains]

Glad it helped!

[@Collingwoodzone]

this is the first time i have properly understood the pdf, the equation and why the integral applies here. excellent explanation - subscribed.

[@iain_explains]

I'm glad you found the video helpful.

[@Carlosbl1996]

With a simple explanation the concept looks way more simple than before. Thanks Iain

[@iain_explains]

Glad you liked it!

[@gayathrigirishnair7405]

Thanks a lot for this simple yet impactful explanation. Probability represented in terms of calculus has always been a roadblock for me. Thanks for helping me in my journey on overcoming this.

[@iain_explains]

Glad it was helpful!

[@jsridhar72]

This is by far the best and easy to understand explanation of PDF. Kudos to the author.

[@iain_explains]

Glad you found it helpful.

[@lg2058]

Must admit this was the best explanation of pdf ive ever heard

[@iain_explains]

Thanks. Glad it was helpful!

[@balinttothg4268]

a 2nd-year physics student trying to understand quantum mechanics is very grateful to you!!!

[@iain_explains]

I'm so glad the video was helpful.

[@rajatguleria4933]

Sir, you really ignited a spark in my mind regarding the need of pdf, I was struggling for long time.

[@iain_explains]

I'm so glad to hear that. Let me know if there are other topics you'd like to hear about (if they're not already on my channel iaincollings.com )

[@sophiacookie437]

The best explanation ever ❤ I finally understand….🎉

[@iain_explains]

Glad it was helpful!

[@VerveLife]

Thank you sir. The best explanation I ever found

[@iain_explains]

Glad it helped.

[@anjangorai2260]

this particular video was more accurate and had a better explanation than the one on the khan academy channel. Thanks a lot for making this concept so simple.

[@iain_explains]

Glad it was helpful!

[@DeadMyth110]

YOU ARE THE FUCKING BEST, idk why the fuck i pay tens of thousands to my university when the tutors just simply read out from the powerpoints word by word and skipping all the steps in between without explaining the logic

[@iain_explains]

Good quality teachers are hard to find. You're right to ask about "value for money" though. I'm glad you like the videos on my channel.

[@sphamandlamay8078]

WoW, That's all I can say. Thanks for the amazing explanation of the concept.

[@LightHolmes]

My professors didn't even bother explaining the notations, added to the confusion about what they were trying to do. Thank you so much!

[@iain_explains]

Yes, that's exactly the experience I had back when I was a student too. It's partly what motivated me to make these videos. I'm glad the video helped. Have you seen my web page with a full listing of all my videos? iaincollings.com

[@LightHolmes]

@@iain_explains Just checked it out, very neat! I know I'll be using it, found so many things I need. Thank you

[@iain_explains]

Gald to help.

[@stevechesney9334]

You're awesome, Sir! Thanks for sharing your expertise and understanding with the community!

[@iain_explains]

Thanks so much. I'm glad you like the videos. I find it rewarding making these videos, especially when I get such nice comments and hear that I'm helping people.

[@hydraofmalice]

You are a god. Why it's so easy and my professor makes out a big deal out of it .

[@iain_explains]

I'm glad you liked my explanation.

[@ALI_B]

I appreciate your approach. It would make many students like the subject instead of disliking it.

[@iain_explains]

Glad you like the videos. Thanks for your nice comment.

[@anoopkumarmishra7056]

Well said.

[@bob-pq6ef]

wow this really made it click in my head, you are an amazing teacher! Have you perhaps thought of doing a series on matrices, I would not be surprised if you know everything there is to know on this as well?

[@iain_explains]

Thanks for your nice comment, and for the suggested new topic. I've added it to my "to do" list.

[@yashs1999]

Amazing explanation, thank you so much! Could never understand prior what f(x) represented, but this video clarified all my doubts.

[@iain_explains]

Thanks. I'm glad it helped!

[@syedrizvi2687]

Amazing explanation, thank you so much for your help! You helped resolve most if not all the questions I had regarding this concept :D

[@iain_explains]

That's great. Glad it was helpful!

[@digipak07]

Greetings from Pakistan. What a simple explanation of seemingly confusing concept. Subscribed.

[@iain_explains]

Glad it was helpful. You might like to check out my other videos on related topics. See www.iaincollings.com for a full list.

[@backare171]

now i wanna learn more! was about to give up till i saw this clear easy explanation..thanks

[@iain_explains]

Glad it was helpful! And I'm glad you're not giving up. Have you seen my other related videos on my webpage? iaincollings.com

[@quickrecipes1638]

Just love your videos. It helped me a lot to clear some of the fundamentals I always struggled with. A big thank you!

[@iain_explains]

You are very welcome! I'm glad you like the videos.

[@AbhimitaDebnath]

Thank you so much for clear explanation! Finally understood PDF!!

[@iain_explains]

Glad it was helpful!

[@MH-ur1jm]

just that what I need, in this manner to understand. thank you

[@iain_explains]

I'm glad it was helpful.

[@fifaham]

Excellent and very simple explanation - thank you Iain.

[@iain_explains]

Glad it was helpful!

[@munyuconeliusgodblessupyou176]

You have perfectly made it for me . Thanks

[@iain_explains]

Glad to hear that.

[@garzottinho]

This is fantastic!!!!! Please keep on posting!!!!!!!

[@iain_explains]

I'm so glad you like the videos.

[@superlinux]

I am now at minute 4:17 of the video. I watched so many videos and searched to understand the physics behind this function and what made people or scientists think of this kind of calculation. Math is meaningless without physics, because math is just the language of physics. The example of traveling is good. Thanks. I will comment as I see something else in this video.

[@iain_explains]

I'm glad you found the example in the video helpful. I agree that physical examples can be extremely helpful in explaining mathematical concepts.

[@superlinux]

@@iain_explains i watched this too kzbin.info/www/bejne/Z4KUo5usftCEpZI It's not from your videos. However, it shows another perspective of finding the formula of f(x) itself graphically. I assure you that you will definitely like it. It will tell you why the function is some kind of an exponential before deriving the formula. It's like a hint. The derivation will have to go through derivatives and integration to get the exact formula.

[@lawrencecohen1619]

Excellent video providing a very intuitive understanding of the probability density function.

[@iain_explains]

Glad it was helpful!

[@harshittalwar1602]

Thanks! This was a lot of help with preparing for my Final.

[@iain_explains]

Glad it helped! I hope your exams went well.

[@NvkChaitanya777]

Wow, It really helps me complete my assignments.

[@iain_explains]

Great. Glad to help.

[@FlemingRound]

What a great explainer!

[@iain_explains]

Thanks. Glad you think so!

[@davidgg8462]

Clear clean simple

[@arandhir]

Wow..... simply amazing explanation 👍👍

[@iain_explains]

Glad you liked it

[@levizwannah]

Thank you. You have clear my confusion of PMF and PDF

[@iain_explains]

Glad to hear that

[@charansai1133]

The explanation is loved it❤

[@iain_explains]

Glad you liked it.

[@kerguule]

Exponential distribution (CDF looks exponential) is a common way to model reliability. If I would now do some Monte Carlo simulation with inverse transform sampling so that I got a group of failure times as an output, do I already see the outcome from that plotted PDF i.e. the higher the Y-axis value peak the more failure times around that time (X-axis) I would expect? The PDF of that exponential function is a decreasing curve but the hazard rate is constant. Why do we call that memoryless (yes, the hazard is constant) even though we would get more failure time values according to the PDF in the beginning (because it looks to be a decreasing curve)?

[@mahmoodmohajer1677]

incredible explanation. thanks for your effort.

[@iain_explains]

Glad you liked it!

[@nash_life]

Very neatly explained.

[@iain_explains]

Glad you liked it

[@chosenbreed]

I think this is a great explanation! Thank you.

[@iain_explains]

Glad it was helpful!

[@Throwingness]

9:00 As a newb, this mostly makes sense, but what do you call a different sliver of the distribution? This dx thing is confusing. You'd name it ox or hx? Any letter instead of x? Not clear to me.

[@iain_explains]

The term dx is a standard term in calculus. It represents a small portion of the "x axis" and is used in integrals and derivatives. The letter "d" is used because it represents "distance" and the "x" indicates that it is a distance in the x-direction.

[@Throwingness]

@@iain_explains Thank you. I was asking about what you would call a different slice of the distribution.

[@eitancohen8717]

Hi, 1:02 - 1:15 "where these are the number of people on that given day" - why given day? you mean "on that given number of hours"? Thanks

[@geministar392]

on the given day where the data is being collected. x is time, y axis is number of ppl

[@iain_explains]

The histogram just shows the people who made the trip on a particular (given) day. On a different day, the histogram would be different, because a different set of people make the trip.

[@pritommazumder9897]

You rock. Keep up the good work!

[@iain_explains]

Thanks! Will do!

[@tank_tales]

This channel is my new Netflix

[@iain_explains]

Great to hear! 😁

[@davidrobins1021]

I've got a question: something's not quite making sense. At 1:27 you've got the diagram showing the journey times of the drivers (presumably rounded to the nearest integer number of hours). You then talk about increasing the precision by reducing the time interval to seconds, microseconds, and so on. As I understand it, as we go to greater precision in this way, there will be a larger number of narrower bars on the diagram. But surely the height of these bars is going to decrease (there will be a certain number of drivers for whom t = 7, say, when rounded to the nearest hour, but hardly any for whom t = 7 hours, 13 minutes and 38 seconds, when rounded to the nearest second). So as we go to infinite precision, where the time increment approaches zero, the smooth curve you draw at 1:53 is not going to go over the bars in the original diagram, but rather will be almost on the horizontal axis. Are you saying that in the limit, as the time increment approaches zero, the histogram becomes essentially a smooth curve, with *non-zero* y-values, and a maximum point that is a specific positive real number? This doesn't seem to make intuitive sense: I'd have thought this "limiting curve" would have a maximum value that is infinitesimally close to zero, as the curve becomes arbitrarily close to coinciding with the x-axis. Are we re-scaling the y-axis in some way? Or have I just completely misunderstood this?

[@iain_explains]

Excellent question. You are right when you say that the height of the bars in the histogram will decrease if we reduce the time interval - but only if we keep the same total number of travellers. However, I said that we were also going to look over every day of the year (but in the limit, it corresponds to "every day, ever"). So we are making the time interval infinitesimally small, but we are also making the total number of travellers become infinitely large. The result is that we are plotting the _density_ of the probability (when we divide by the total number of travellers). If you want to know the _exact_ probability, you need to integrate the density over a range of values. If the range of values is zero (ie. you are asking about an _exact_ value), then the integral will be zero (and hence the probability will be zero). As I point out at the 6:30min mark, the probability of any _exact_ value of a continuous random variable, equals zero. But this doesn't mean that the probability _density_ is zero (or even infinitesimally small).

[@davidrobins1021]

@@iain_explains Thanks for the reply. OK that makes more sense now; I guess I was missing the distinction between probability density and probability itself.

[@twobunnyrabbit]

Thanks for the excellent explanation. Now I know why dx is there! And what f(x)dx mean. And what that integral sign means.

[@iain_explains]

I''m glad the video helped.

[@farnandomadera8299]

Hello sir, nice explanation of the concept of the PDF but i think that there is a little forget in a formula in minute 8:33, when explain the height of the density you forget to write the integral sign and the limits of the integral might be x to x+dx.

[@iain_explains]

No, that's not the case. The expression in the video is correct. Perhaps you are missing seeing the "dx" term? That is the width/distance between x and x+dx. Since the "height" of the pdf is constant over that region, there is no need to write the integral sign from x to x+dx. The value of that integral is simply the base times the height (where dx is the base).

[@abdullahraihanbhuiyan2346]

Very nice and to the point!

[@iain_explains]

Glad you liked it.

[@aswaths4184]

Thanks a lot for this

[@iain_explains]

My pleasure

[@AJ-et3vf]

awesome video sir! Thank you!

[@iain_explains]

Glad you liked it!

[@infj4w511]

Do you have anything on return plots?

[@iain_explains]

Not at the moment. But thanks for the suggestion, I've added it to my "to do" list.

[@JitendraNathNaik-p9u]

Thank you sir. ❤

[@iain_explains]

You're welcome 😊

[@wilsongomes3360]

Very good explanation

[@iain_explains]

Thanks. I'm glad you liked it.

[@nak6608]

Excellent video thank you very much!

[@iain_explains]

Glad it was helpful!

[@kin_1997]

amazing explanation

[@iain_explains]

Glad you liked it

[@dempsey3810]

Nice video. Explains the basic concept of PDF well. However, kind of a shame that it stops before demonstrating a specific case study.

[@iain_explains]

If you check out my webpage you'll find a few videos with examples of PDFs, including Gaussian, Chi-Square, ... iaincollings.com

[@jamesb5213]

This is slightly related to this, but Im taking a class on random variables. We encountered the "density method" for finding the pdf of a function of a random variable, as in using the derivative of the inverse to find the new pdf. My question is, why do we only use the first derivative to change between pdfs? As in, what intuition lies behind the fact that you only need the linear relationship between the two variables to change between pdfs?

[@iain_explains]

Great question. It's hard to explain in these comments. I think I'll have to make a video on this topic.

[@Dansel_B]

hey Iain, thanks for the video. I've got a question though. my take is that you come from PDF to CDF (by calculating of the probability of the area of the range), did I get it right?

[@iain_explains]

You get the CDF by integrating (taking the area under) the PDF from negative infinity to the value, x.

[@chenwu7667]

Thanks

[@wanassaben8269]

Thank you 🧠💛🇩🇿

[@iain_explains]

You're welcome.

[@youyangpeng9710]

Thank you so much for the teaching. Just i don't understand why the total of the probability is the whole area instead of the function of all variables adding up.

[@iain_explains]

Sorry, I don't know what you mean by: "the function of all variables adding up"

[@rdanimetalks7964]

Sir Plzzz Help me I just have one Doubt .... In my book the probability function is Mentioned as f(x, theta)...do what's the Theta in this one

[@iain_explains]

In that notation, the "theta" refers to a constant (non-random) parameter in the function.

[@cardflopper3307]

is it correct to say that the output of a (continuous) PDF is a density rather than a probability. ie The word "density" in Probability Density Function is more imporant to emphasize? I've always struggled with what exactly the PDF represents. What has helped me (i hope) is seeing the connection of the calculation of Center of Mass (of an object) to the Mean (of random variable). Also Variance --> Moment of Inertia

[@iain_explains]

What do you mean by "the output of a (continuous) PDF"? What do you mean _output_ ? In terms of your comment about the word "density", I think a good analogy is to think of a gas which has a certain density of particles. If you have a particular volume (eg. a balloon), then you can calculate the mass of the gas in that balloon (density times volume), but if you only have a specific point in space that you're interested in, then it's impossible to say how much gas is at that exact point. So for continuous-valued random variables, it's not possible to plot the exact probability for an exact point/value, all you can do is plot the density.

[@EigenA]

Well done 👍

[@iain_explains]

Thank you 👍

[@highereducation1444]

Thnk u sir because I have done two days to understand that topics but still not understood to myself

[@iain_explains]

I'm glad to hear that my video was helpful to you.

[@eitancohen8717]

Do you also explain Gamma and Beta distributions? Or are you intend to do that? Again, thanks a lot

[@iain_explains]

Thanks for the suggestion. I've put them on my "to do" list.

[@Impillion]

Great video sir

[@iain_explains]

Glad you liked it.

[@mnada72]

Is there a prerequisite courses for this topic ? If so please suggest some.

[@iain_explains]

I'd suggest watching my Probability and Random Processes playlist on my channel. It starts with this video: "What is a Random Variable?" kzbin.info/www/bejne/g36ZgoBprp2jrKs

[@sumopuppy1248]

Would you be able to do a video on more general notation consideration in mathematics? Is there any fundamental mathematical principle that drives using a capital X vs a small x? Or this is just random and a matter of style?

[@iain_explains]

It's traditional to use capital letters to represent random variables, and lower case letters to represent specific values that the random variable can take.

[@anoopkumarmishra7056]

@@iain_explains Thank you, Sir!!

[@naurinrisat71]

Respect boss!

[@tilakvarisety7711]

Thanks for the video. Still, cant quite understand why the probability of obtaining a RV X is Zero, and how wide dx should be? It is Zero because its a continuous RV? Would it change for a discrete RV?

[@iain_explains]

Yes, exactly. Because it is a continuous valued RV, the probability of any one _exact_ value is zero (because there are an infinite number of possible _exact_ values, infinitely closely spaced). But it is possible to define a probability _density_, which is what the p.d.f. is. And yes, for discrete valued RVs, there are distinct gaps between the values, and as such, it is possible to define exact probabilities in that case.

[@rakeshkulkarni4323]

very good explaination but only one question why do we take area ?

[@iain_explains]

Sorry, I don't know what you're asking. The PDF is a density function (as the name indicates). So to find out the probability of an event, it is necessary to add up the probability density, over the range of that event.

[@rakeshkulkarni4323]

@@iain_explains got it thank you

[@donibrasco1984]

Wow. Awesome. Thank youuuuuuu

[@iain_explains]

Glad you liked it!

[@donibrasco1984]

Can you help me understand what would be the height. Like what wud it mean if height is fx (x)?

[@iain_explains]

The name gives the clue. The height is the "density" of probability - as in "probability per unit value". Think about a gas in a jar with a density of particles per unit volume. The total amount of gas is the density times the volume. For probability, the total probability for a certain range of "outcome values" is the density of the probability times the range of the outcome values.

[@donibrasco1984]

@@iain_explains I got it. It's length of a rectangle of a bar of a histogram that when gets multiplied with base gives total area. And u have put it a better way by saying it's per unit probability. And on base we have units in a range form. Thanks alot.

[@canocan5050]

Can you do a transformation of random variables using the pdf method where Y is 0.5*cos(pi*X) - 0.5 and X is uniformly distributed from [0,4]. The task is to write and draw the pdf of the output Y. Please explain all the steps and just how the new intervals result.

[@iain_explains]

Thanks for the suggestion. I've put it on my "to do" list.

[@petermahoney4518]

Thanks very much for this. I am having trouble with this idea: If the x axis has infinite precision and this makes it impossible to assign a probability to an exact value of x, why doesn't this also make it impossible to say precisely what the bounds of the integration are? Is it that, while there are infinitely many x values, this doesn't make it impossible to land on one exactly, at least theoretically? Thanks again.

[@iain_explains]

I think a good analogy is to think of a gas which has a certain density of particles. If you have a particular volume (eg. a balloon), then you can calculate the mass of the gas in that balloon (density times volume), but if you only have a specific point in space that you're interested in, then it's impossible to say how much gas is at that exact point.

[@hannesjakob2041]

Bless you man

[@iain_explains]

Thanks. Glad you found it helpful. You can find a listing of other related videos on the channel at: www.iaincollings.com/probability-and-random-variables

[@গোলামমোস্তফা-শ৮থ]

But why you take the range minus Infinity to plus Infinity?

[@iain_explains]

Because that is the range of values that the random variable can take (in general). If you have a RV with a finite range of possible values, then you only need to integrate over those values.

[@brngylni]

Sir, it is very good but how can it be greater than 1? I have seen that it can be greater than 1 when the interval is too small.

[@iain_explains]

The thing to remember is that it is a probability _density_ function - not the probability. You get the probability by taking the area under the curve. As I said in the video, the actual probability of any _exact_ value is zero (for continuous valued RVs) since the "range" (range of integration) of an _exact_ value is zero (infinitely narrow). Therefore the pdf can have any height, as long as the total area under the curve equals 1.

[@anon4286]

thank you so much!

[@iain_explains]

You're welcome!

[@ilhsillhs2239]

super. thank you

[@iain_explains]

Glad you liked it.

[@ronwood3886]

Thanks you!

[@iain_explains]

You're welcome!

[@lokeshprofessional4647]

awesome explaination sir.. i had doubt....number of person on travels (losangles to sanfracasco)within 5hours is 20person.....its % =(number/total)*100----->means here is number=20,total =5hours...we will get as (20/5)*100=400%........what is number?here...what is total...mean total hours or total means total numbers of persons....

[@iain_explains]

Sorry, I'm not sure what you're asking exactly. I'm not showing the number of people who traveled "within 5hours", I'm showing the number of people who took 5 hours to make the journey.

[@bubblez9266]

wow, it is so easy....

[@iain_explains]

Glad you liked the explanation.

[@abhishektakeineasy1882]

Why fxdx gives us the probability

[@iain_explains]

As I say in the video, f(x) is the probability density (not the probability), so it needs to be multiplied by the range of values that you're interested in. It's analogous to working out the amount of gas in a balloon, by multiplying the density of the gas by the volume of the balloon.

[@renatoberaldo2335]

Where is the square of amplitude?

[@iain_explains]

You’ll need to give the context. Otherwise I have no idea what your question refers to.

[@jobalvaro3440]

Nice presentation but voice is quite muffled..

[@iain_explains]

Sorry for that

[@mohammedghouse235]

Well, I'd say that you are putting in lot of effort into making these videos. Keep up the good work! However, I'd also like to suggest that your videos are difficult to follow. I mean the explanations are a bit difficult. Even the simple concepts you make them seem complicated.

[@iain_explains]

Hmm, that's interesting feedback. It's basically the total opposite to what almost everyone else says about the videos. Perhaps my explanations of the simple concepts seem complicated to you because you alreay know all about them, and think they don't need extra explanation. Or perhaps you think they are more simple than they actually are - which is why I add the extra detail to try to point out the complexities and subtelties. Anyway, I will give your comment some more thought. I am always trying to make sure my videos are clear to follow.

[@mohammedghouse235]

@@iain_explains Thank you, Iain. I was just being supportive. :) . And yes. adding more details is good, but for people who already know about the concept. On the other hand, you could make a simpler version and a detailed version separately. So, that viewers can get started with the simpler one instead of being blown away by the complexities of the concept being discussed. Cheers!

[@alvegrande7398]

damnnn ur mind thank u so much for sharing

[@iain_explains]

It's great to hear that you liked the video and found it useful.

[@Stellectis2014]

5 hours? That's 77.8MPH. not possible there's jerks who drive 65MPH in the fast lane. Better analogy would be a helicopter ride but the wind varies in certain regions. Thank you for the explanation. You could use an example of L.A to Santa Monica. you use the average speed of 2MPH and contrast the people walking will get there sooner jogging.

[@nonyafletcher601]

So bad. No one ever explains how the raw data transforms into the probability density function. Or when your distribution is not a bell curve. Why is this so?

[@iain_explains]

Sorry, I don't understand your question. I did explain how the raw data from an "experiment" relates to the pdf. And yes, it doesn't have to be a bell curve. Here are some videos of non-bell curve examples: "What is Rayleigh Fading?" kzbin.info/www/bejne/Y3eyn4x4j5yNnLM and "What is Rician Fading?" kzbin.info/www/bejne/h5rWk4xpbM2qnqs

[@Dina-he1uc]

😄😄😄