Related to the above question, what does $p(r, \theta) dr d\theta$ represent?
OillyNoodle
I think the 2nd line comes from the first one. Treating the total area as 1 (divided by pi), we get the p(r, theta)
KrystalTea
To my understanding, the 2nd formula sees the shaded area as a rectangle (since dr is very small). Then the area of this small rectangle becomes length (rd\theta) * width (dr). Plus the area of the circle is pi. Please correct me if I'm wrong.
supernova
I think the 2nd formula is just the unit area / the whole circle area.
ilovecg
http://15462.courses.cs.cmu.edu/fall2019/lecture/numericalintegration/slide_034 might be helpful.
qiqinl
For a distribution w.r.t. x and y, say $f(x,y)$, if you perform coordinate transformation(to polar coordinate), since $f(x,y)dxdy = g(\theta,r)rd\theta dr$, you got the distribution w.r.t. $\theta$ and r which is $g(\theta,r)r$, where $g(\theta,r) := f(rcos\theta, rsin\theta)$.
hesper
You guys may want to check out the Box-Muller Algorithm :)
https://en.wikipedia.org/wiki/Box–Muller_transform
How does the 2nd formula comes?
Related to the above question, what does $p(r, \theta) dr d\theta$ represent?
I think the 2nd line comes from the first one. Treating the total area as 1 (divided by pi), we get the p(r, theta)
To my understanding, the 2nd formula sees the shaded area as a rectangle (since dr is very small). Then the area of this small rectangle becomes length (rd\theta) * width (dr). Plus the area of the circle is pi. Please correct me if I'm wrong.
I think the 2nd formula is just the unit area / the whole circle area.
http://15462.courses.cs.cmu.edu/fall2019/lecture/numericalintegration/slide_034 might be helpful.
For a distribution w.r.t. x and y, say $f(x,y)$, if you perform coordinate transformation(to polar coordinate), since $f(x,y)dxdy = g(\theta,r)rd\theta dr$, you got the distribution w.r.t. $\theta$ and r which is $g(\theta,r)r$, where $g(\theta,r) := f(rcos\theta, rsin\theta)$.
You guys may want to check out the Box-Muller Algorithm :) https://en.wikipedia.org/wiki/Box–Muller_transform