How to Generate Random Points Inside a Circle

⭐Rejection sampling is a method that randomly selects points inside a circle by rejecting points that fall outside the circle.

🔑Inverse transform sampling is another method that transforms a uniform distribution to achieve a desired distribution, such as a linear distribution inside a circle.

📐A circle can be thought of as a collection of infinitely thin isosceles triangles. By selecting a random triangle and point inside the triangle, a uniform random point inside the circle can be obtained.

🧮The probability of selecting a point inside a circle using rejection sampling is equal to the area of the circle divided by the area of the square that bounds the circle.

🎲Python code examples are provided to implement the rejection sampling and inverse transform sampling methods.

What is rejection sampling?

—Rejection sampling is a method that randomly selects points from a distribution by rejecting points that fall outside a desired range or shape.

How does inverse transform sampling work?

—Inverse transform sampling is a method that transforms a uniform distribution to achieve a desired distribution by inverting the cumulative distribution function (CDF).

How can random points be generated inside a circle?

—Random points can be generated inside a circle using rejection sampling, where points falling outside the circle are rejected, or using inverse transform sampling, which transforms a uniform distribution to achieve a linear distribution inside the circle.

What are the advantages of rejection sampling?

—Rejection sampling is a simple method that can be easily implemented and works well for shapes with simple boundaries, such as circles.

How can I implement these methods in Python?

—Python code examples are provided in the video to demonstrate the implementation of rejection sampling and inverse transform sampling to generate random points inside a circle. Simply follow along with the code examples to apply these methods in Python.

00:00Introduction and motivation for generating random points inside a circle.

04:14Explanation of rejection sampling method for generating random points inside a circle.

09:58Explanation of inverse transform sampling method for generating random points inside a circle.

11:11Explanation of using isosceles triangles for generating random points inside a circle.

12:50Demonstration of Python code implementation for both rejection sampling and inverse transform sampling methods.