Normalising the Normal Distribution #maths #calculus #statistics

The Normal Distribution is one of the most important objects in Statistics; it can model all kinds of things in nature and society, such as the wavefunction of a particle, the height of students in a class... Th Gaussian function is used as the Probability Density Function for the Normal Distribution, and the main reasons include: it has a peak at the centre, decays to zero very quickly, is smooth everywhere...
This video showed how to move from the simplest Gaussian integral to the
Normal Distribution PDF with its parameters.
The Gaussian integral is very well-known, you can just search up in the internet on how to calculate it (a lot of methods exist)
Key points during the transformation:
x-mu moves the peak of the function to x=mu
v(x-mu) scales the function such that the standard deviation can be accounted (where one standard deviation away corresponds to the inflection point of the function)
The sqrt(pi) in the denominator makes sure that probabilities add up to 1
(Apart from using a calculator, the last integral can also be calculated using integration by parts. This is done by letting u=u and dv=u*e^(-u^2)*du)

Видео Normalising the Normal Distribution #maths #calculus #statistics канала The Amateur Tutor