Excellent explanation. Great job. I am particularly impressed that you explained even the minutest details. Thanks. Can you post similar videos for other distributions as well. For example, Beta, Weibull, Poison etc.
@@molloymaths1092 Hi, I was presented this density function in my finance course as a part of the Black-Scholes formula (which I guess you have knowledge of). Im trying to grasp where this actually comes from and your video has helped a lot. However, I have a couple of questions and would be grateful if you could try to help me out. 1. How can you just make the assumption that y is 0 and what kind of implications does this really have? I mean, if y is 0 then graphically you just have an x-value going right and left? 2. How do you actually know that you can change g(x) to a function of the form e^kx^2. Is this a result of some kind of trial and error? I do know that e is "important" but why for example dont you try a function like g(x) = 25 - 3x/4 + x^2 ? To my lack of mathematical insight it seems like you took it out of thin air. Thank you and have a nice day! Best regards, Johanna
@@johannaw2031 Hi. Thanks for your comment. You can let y=0 since you are doing it on both sides of the equation. Also it doesn't change the probabilities as r doesn't change. There is a certain amount of trial and error as well as intuition when substituting e^kx^2. It does work as I have shown that in the video. Thanks again for your comment.
I wonder if there are variants of the normal distribution, by this I mean continious functions who satisfy the original assumptions and ehen integrated from neg inf to pos inf equals one. This could be explored by taking the line g(x)g(y)=g(sqrt(x^2 + y^2)) and trying to find other solutions for g. Or maybe it can be proved the only a certain family kf functions exist, idk.
Haven't seen any video or full proper explanation about this derivation, from a student in need for his school main project thanks alot and im definitely sourcing you for my project(don't worry lol).
thank you so much. i remember watching this a while ago and now I needed it. happy you had it on youtube still. I'll prefer to fix k = -1 first to make the derivation easier. it follows sigma^2 = 1/2. and from that we can apply lineartransformations.
Quick question. After rewriting the function based on f(y) where y=0 is a constant, everything else flows smoothly. I dont understand, however, how y itself can be regarded as static without the derivation restricting calculations of other y-values. Do let me know if you would like me to clarify my question. Thank you.
Safe to say I couldnt understand anything past 9:00. Could you leave a trail of references and math topics that a student could look into to better understand?
Thanks. The probability of the dart landing in the x direction is independent of the probability of the dart landing in the y direction. When two events are independent you multiply their probabilities.
Fantastic Video!!! I love the way you make the explanation! A quick question at 31:10 I don't quite understand why you consider x goes to infinity and the red-circle-part becomes null. Surely if we plug in 0, it is 0 as well. But if we plug in 1, it has value not equal to 0. Thanks!
Enjoying the video! But at 10:43 when you derive that [2]: w(x) = L*f(x), we can pair this nicely with the fact directly above [1]: w[sqrt(x^2 + y^2)] = f(x) * f(y) to yield: By substituting [2] into the LHS of [1] L * f(sqrt(x^2 + y^2)) = f(x) * f(y) We can then divide both sides by L^2 to obtain the result you achieve around 12:26 L* f(sqrt(x^2 + y^2)) / L^2 = f(x) * f(y) / L^2 => f(sqrt(x^2 + y^2)) / L = f(x) / L * f(y) / L I think this step would probably have been a lot quicker than what you used, but perhaps there is a reason you followed the steps you did? Again, enjoying the vid!
I think that there are a number of ways of getting through this derivation. I have just given one version. Thanks for watching and for your interesting comment.
@@molloymaths1092 were there any assumptions made about the spread of darts initially? And would other functions exist that satisfy the functional equation you derived? Ones that would produce other, just as valid probability density functions?
@@madmorto2610 I didn't make any assumptions about the spread of the darts. Sorry but I'm not sure of any other functions that can be derived from the functional equation.
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Excellent explanation. Great job. I am particularly impressed that you explained even the minutest details. Thanks. Can you post similar videos for other distributions as well. For example, Beta, Weibull, Poison etc.
Thanks. Haven't done those yet but will have a look at doing them at some stage.
Great video! I particularly liked how you took the time to highlight the relationship between lambda and the spread.
Thanks. Glad you liked it.
Brilliant explanation! Love how you took care of any questions we would potentially in between the steps. Subscribed! 🙏
Thanks and your welcome!
Very great video sir! I am very happy get your explanation about deriving the pdf of normal distribution
You're welcome
Will be helpful for those want to know deep about the topic.
Excellent video, especially when you piece it together with derivation of the Gaussian Integral. Thank you very much!
Your welcome. Glad you liked it.
Helpful. Thank you for going through every step.
Your welcome.
@@molloymaths1092 Hi,
I was presented this density function in my finance course as a part of the Black-Scholes formula (which I guess you have knowledge of). Im trying to grasp where this actually comes from and your video has helped a lot.
However, I have a couple of questions and would be grateful if you could try to help me out.
1. How can you just make the assumption that y is 0 and what kind of implications does this really have? I mean, if y is 0 then graphically you just have an x-value going right and left?
2. How do you actually know that you can change g(x) to a function of the form e^kx^2. Is this a result of some kind of trial and error? I do know that e is "important" but why for example dont you try a function like g(x) = 25 - 3x/4 + x^2 ? To my lack of mathematical insight it seems like you took it out of thin air.
Thank you and have a nice day!
Best regards,
Johanna
@@johannaw2031 Hi. Thanks for your comment.
You can let y=0 since you are doing it on both sides of the equation. Also it doesn't change the probabilities as r doesn't change.
There is a certain amount of trial and error as well as intuition when substituting e^kx^2. It does work as I have shown that in the video.
Thanks again for your comment.
I wonder if there are variants of the normal distribution, by this I mean continious functions who satisfy the original assumptions and ehen integrated from neg inf to pos inf equals one.
This could be explored by taking the line g(x)g(y)=g(sqrt(x^2 + y^2)) and trying to find other solutions for g. Or maybe it can be proved the only a certain family kf functions exist, idk.
Haven't seen any video or full proper explanation about this derivation, from a student in need for his school main project thanks alot and im definitely sourcing you for my project(don't worry lol).
Great. Glad it helps. Good luck with your project.
thank you so much. i remember watching this a while ago and now I needed it. happy you had it on youtube still.
I'll prefer to fix k = -1 first to make the derivation easier. it follows sigma^2 = 1/2. and from that we can apply lineartransformations.
Your welcome. Glad it helps.
Exactly what I am seeking . Thanks
Great. Glad it helped.
Thank you very much,perfect explanation,and now I can feel comfortable...
Thanks. Glad it helps.
Outstanding, clear and precise. How about if you add examples to the use of the CDF? With many thanks and appreciations
Thanks for your comment.
Excellent explanation Sir👏👏, thank you
Your welcome!
Quick question. After rewriting the function based on f(y) where y=0 is a constant, everything else flows smoothly. I dont understand, however, how y itself can be regarded as static without the derivation restricting calculations of other y-values. Do let me know if you would like me to clarify my question. Thank you.
Not sur exactly what you mean!
Safe to say I couldnt understand anything past 9:00. Could you leave a trail of references and math topics that a student could look into to better understand?
You'll have to be more specific.
A very elegant video. A quick question, why does w(r) = f(x).f(y) ?
Thanks. The probability of the dart landing in the x direction is independent of the probability of the dart landing in the y direction. When two events are independent you multiply their probabilities.
Fantastic Video!!! I love the way you make the explanation! A quick question at 31:10 I don't quite understand why you consider x goes to infinity and the red-circle-part becomes null. Surely if we plug in 0, it is 0 as well. But if we plug in 1, it has value not equal to 0. Thanks!
e to the power of minus (Pi lambda squared x squared) is 1/(Pi lambda squared x squared) which becomes zero if x goes to infinity.
@@molloymaths1092 i understand its the denominator. But why would x goes to infinity in this case?
@@hiuyingchoy5399 We are integrating from +/- infinity
Enjoying the video! But at 10:43 when you derive that [2]: w(x) = L*f(x), we can pair this nicely with the fact directly above
[1]: w[sqrt(x^2 + y^2)] = f(x) * f(y) to yield:
By substituting [2] into the LHS of [1]
L * f(sqrt(x^2 + y^2)) = f(x) * f(y)
We can then divide both sides by L^2 to obtain the result you achieve around 12:26
L* f(sqrt(x^2 + y^2)) / L^2 = f(x) * f(y) / L^2
=> f(sqrt(x^2 + y^2)) / L = f(x) / L * f(y) / L
I think this step would probably have been a lot quicker than what you used, but perhaps there is a reason you followed the steps you did? Again, enjoying the vid!
I think that there are a number of ways of getting through this derivation. I have just given one version. Thanks for watching and for your interesting comment.
Excellent video!
Thanks!
@@molloymaths1092 were there any assumptions made about the spread of darts initially?
And would other functions exist that satisfy the functional equation you derived? Ones that would produce other, just as valid probability density functions?
@@madmorto2610 I didn't make any assumptions about the spread of the darts. Sorry but I'm not sure of any other functions that can be derived from the functional equation.
@@molloymaths1092 ok, I think there is an implicit assumption made about the spread of the darts when choosing to use exp function.
Excellent and thank you so much
Your welcome.
What is the symbol you call "ORR" or "OHR" that looks like an r? Is this a greek letter ?
It's just an r
I couldn't understand
If w(x)=lambda.f(x) only when y=0 then after that all the steps involving y are just identities like 1=1
Explain?
Where?
Do you multiply f(x) with f(y) because you want to express AND?
yes.
Thank you!
Your welcome!
Can you please tell the reference?
Not sure what you mean
@@molloymaths1092 If you have learned this from a specific book, tell me its name if you could remember. thank you for replying.