Yes and no. The error function erf(x) is slightly different than the integral of the normal distribution curve. It is the integral of the normal distribution curve, but with a scaling constant and an asymptote at y=-1 and y=+1, that is set up this way for its applications in differential equations. For the CDF of the normal distribution, we want asymptotes at 0 and +1, so the function is shifted and scaled. The integral of the standard normal distribution curve in terms of erf(x) is as follows: integral Z(x) dx from -infinity to X = 1/2*erf(X/sqrt(2)) + 1/2 And the integral of the base form of this function, e^(-x^2), in terms of erf is: 1/2*sqrt(pi)*erf(x) + C
There is a way to do the full domain integral of the bell curve without infinite series, that involves squaring it, generating a 3-D bell curve, and transforming it to polar coordinates to carry it out. The coordinate transformation turns dx dy into r dr dtheta. This generates r*e^(-r^2) as the integrand, which we can solve with simple substitution. The volume of the 3D bell curve is pi, and sqrt(pi) becomes the area of the original 2-d bell curve.
When it is a power series (each term is x^n multiplied by some number that doesn't depend on x, just on n) then you can always do that. Also interchanging derivative and summation.
I think maybe I understand, you would integrate every term of the summation and sum the resulting integrals, and these are actually the terms of the new summation on the solution
Yes the method is like that, but he forget to specify that the series converges uniformly in [0,1]. Without this hypothesis you can’t switch the integral sign with the summation sign. {Sorry for the answer after a year ;)}
i love you. i appreciate this video more than you can ever understand. if you ever need anything, let me know.
excellent video. in advanced math though, im struggling to understand a lot of weird taylor expansions that pop up from nowhere :)
Sometimes you just need to accept the Taylor series, no matter where it appears 😅
Thank you, Professor. Your examples are very nice to have when I'm tutoring students.
Very helpful in solving problems of iit entrance (jee advance).
Love the man’s Gusto, but it would be a lot better if he wrote it out as he was talking.
Great video ... but I always understood that the integral of the normal distribution curve does have a name "The Error Function" ?
Yes and no. The error function erf(x) is slightly different than the integral of the normal distribution curve. It is the integral of the normal distribution curve, but with a scaling constant and an asymptote at y=-1 and y=+1, that is set up this way for its applications in differential equations. For the CDF of the normal distribution, we want asymptotes at 0 and +1, so the function is shifted and scaled.
The integral of the standard normal distribution curve in terms of erf(x) is as follows:
integral Z(x) dx from -infinity to X = 1/2*erf(X/sqrt(2)) + 1/2
And the integral of the base form of this function, e^(-x^2), in terms of erf is:
1/2*sqrt(pi)*erf(x) + C
Hi Trefor, very helpful video! One question, for your example on limits at 5:00, why do the larger powers go to zero faster?
@@DrTrefor Thank you! Cheers.
Woah 🤯 he has a huge chalkboard
Awesome video! perfect explanation of why in the heck I learned Taylor Series
Great
What does integrating integrate?
Thanks for the video.
Thank you, now what if the upper limit of the integral is infinity? Could you still use Taylor series to solve it?
yes, of coure!
There is a way to do the full domain integral of the bell curve without infinite series, that involves squaring it, generating a 3-D bell curve, and transforming it to polar coordinates to carry it out. The coordinate transformation turns dx dy into r dr dtheta. This generates r*e^(-r^2) as the integrand, which we can solve with simple substitution. The volume of the 3D bell curve is pi, and sqrt(pi) becomes the area of the original 2-d bell curve.
Taylor series are actually goated for doing tough limits.
Also wanted to thank you, great video.
Awesome..
Excellent video. Last part was explained beautifully
Sir when are we allowed to interchange and integral and summation?
When it is a power series (each term is x^n multiplied by some number that doesn't depend on x, just on n) then you can always do that. Also interchanging derivative and summation.
I don't get how you integrated the general term
Neverrmind I didn't realize n was a constant
Application of mechlerun series are Similar to taylor??
Yes it just when u centre around 0
Landau symbols????
Taylor Swift or Taylor Series?
If you type "taylor" into any search bar, she shows up first.
"Taylor" of the Taylor Series is Brook Taylor.
can you please make a video on how to integrate a summation? I got a little lost at 2:08
I think maybe I understand, you would integrate every term of the summation and sum the resulting integrals, and these are actually the terms of the new summation on the solution
Yes the method is like that, but he forget to specify that the series converges uniformly in [0,1]. Without this hypothesis you can’t switch the integral sign with the summation sign. {Sorry for the answer after a year ;)}
@@franzmaina3080 thanks!
from india
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The best explanation I've seen.
I was expecting you to give the disclaimer that swapping the integral and summation isn't always allowed but it is in this case
could you please tell me why
wrong, you must use the symbols of landau
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Why didn't you use the basic formula of power series in evaluating integral of e^-x^2, the derivative of e^-x^2 will change
Mr. Bazett, Tarzan speaks better than you, that is, slowly, clearly and finally in an understandable way. Cheers, yop.