Different types of distributions can really change how the Law of Large Numbers (LLN) works in real life. Here are some thoughts based on what I've seen:

1. Types of Distributions:

   • Normal Distribution: When you're dealing with normally distributed data, the LLN works really well. As you gather more data, your average quickly gets closer to the true average.
   • Exponential Distribution: In this case, the average is clear, but if you're waiting for rare events (like a bus), it might take longer to see things even out.
   • Heavy-tailed Distributions: An example is the Cauchy distribution, which can complicate things. Here, the average doesn’t even exist, making it hard to use the law.

2. Speed of Convergence:

   • Distributions with a definite (finite) variance, like the normal distribution, let you get to the expected value faster than those with an indefinite (infinite) variance.

3. Practical Implications:

   • If you’re doing experiments or simulations, knowing what type of distribution you have can help you figure out how quickly your averages will settle down around the true average.

In summary, how well the Law of Large Numbers works really depends on the type of distribution you’re using. This is something important for statisticians to remember!