Link to original video by math et al

Inverse Transform Sampling | Triangular Distribution

Inverse Transform Sampling: Triangular Distribution

Short Summary:

• This video demonstrates the inverse transform sampling method, a technique for generating random samples from a desired probability distribution.
• The video focuses on a specific example: generating samples from a triangular distribution using a uniformly distributed random variable.
• The process involves converting the probability density function (PDF) into a cumulative distribution function (CDF), finding its inverse, and then plugging in a uniformly distributed random variable.
• This method allows for the generation of samples from complex distributions using a simple uniform distribution.

Detailed Summary:

1. Introduction:

• The video begins by introducing the concept of inverse transform sampling as a method for generating random samples from a desired distribution.
• It mentions a previous video on the topic and provides a link for further reference.
• The specific example used in this video is to generate samples from a triangular distribution, a distribution with a PDF resembling a triangle.

2. Steps of Inverse Transform Sampling:

• The video outlines the three main steps involved in inverse transform sampling:
  • Step 1: Find the CDF: Convert the given PDF into its corresponding CDF.
  • Step 2: Find the Inverse CDF: Determine the inverse of the CDF.
  • Step 3: Plug in Uniform Variable: Substitute a uniformly distributed random variable (U) into the inverse CDF.
• This process generates samples from the target distribution.

3. Finding the CDF:

• The video demonstrates finding the CDF for the triangular distribution, which involves integrating the PDF over different intervals.
• The CDF is defined in two parts, corresponding to the two segments of the triangular distribution.

4. Finding the Inverse CDF:

• The video then focuses on finding the inverse of the CDF, which also involves two parts.
• For each part, the video solves for X in the equation Y = F(X), where Y represents the CDF and X represents the random variable.
• The video highlights the importance of using the infimum definition to ensure a unique solution for the inverse CDF.

5. Generating Samples:

• The final step involves plugging in a uniformly distributed random variable (U) into the inverse CDF.
• This generates samples from the target triangular distribution.

6. Conclusion:

• The video concludes by summarizing the process of inverse transform sampling and emphasizing its usefulness for generating samples from complex distributions.
• It encourages viewers to explore the topic further and provides links to additional resources.