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 ;)}
Thank you, Professor. Your examples are very nice to have when I'm tutoring students.
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 😅
Very helpful in solving problems of iit entrance (jee advance).
Awesome video! perfect explanation of why in the heck I learned Taylor Series
Love the man’s Gusto, but it would be a lot better if he wrote it out as he was talking.
Excellent video. Last part was explained beautifully
The best explanation I've seen.
Also wanted to thank you, great video.
Taylor series are actually goated for doing tough limits.
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
Thanks for the video.
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.
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.
Woah 🤯 he has a huge chalkboard
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.
Awesome..
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!
I don't get how you integrated the general term
Neverrmind I didn't realize n was a constant
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
Great
Landau symbols????
Application of mechlerun series are Similar to taylor??
Yes it just when u centre around 0
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
from india
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.
Help
wrong, you must use the symbols of landau
講嘢唔駛咁大動作嘛
🌍
Mr. Bazett, Tarzan speaks better than you, that is, slowly, clearly and finally in an understandable way. Cheers, yop.