🧵1/10 - Law of Large Numbers (LLN) in R 📈

Hello #Rstats community! Today, we're going to explore the Law of Large Numbers (LLN), a fundamental concept in probability theory, and how to demonstrate it using R. Get ready for some code! 🚀

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🧵2/10 - What is LLN? 🧐

LLN states that as the number of trials (n) in a random experiment increases, the average of the outcomes converges to the expected value. In other words, the more we repeat an experiment, the closer we get to the true probability.

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🧵3/10 - Coin Flip Example 🪙

Imagine flipping a fair coin. The probability of getting heads (H) is 0.5. As we increase the number of flips, the proportion of H should approach 0.5. Let's see this in action with R!

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🧵4/10 - R: Coin Flip Simulation 📊

First, we'll create a simple function to simulate a coin flip:

flip_coin <- function(n) {
outcomes <- sample(c("H", "T"), n, replace = TRUE)
return(outcomes)
}

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🧵5/10 - R: Running the Experiment 🧪

Now, let's run the experiment for different numbers of flips & store the proportion of H:

n_trials <- seq(10, 100000, 5)
proportion_heads <- sapply(n_trials, function(n) {
flips <- flip_coin(n)
mean(flips == "H")
})
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🧵6/10 - R: Visualizing the Results 📉

We'll use ggplot2 to visualize the results:

library(ggplot2)

data <- data.frame(n_trials, proportion_heads)
ggplot(data, aes(x = n_trials, y = proportion_heads)) +
geom_line() +
geom_hline(yintercept = 0.5, linetype = "dashed") +

ylim(c(0,1)) +
scale_x_log10() +
labs(title = "Law of Large Numbers - Coin Flips",
x = "Number of Trials",
y = "Proportion of Heads")

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🧵7/10 - Interpreting the Plot 📈

The plot shows that as the number of trials increases, the proportion of H converges to 0.5, confirming the Law of Large Numbers.

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🧵8/10 - Real-World Applications 🌎

LLN has many real-world applications, such as predicting stock prices, insurance premium calculations, and assessing the performance of machine learning models.

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🧵9/10 - Limitations of LLN ⚠️

Keep in mind that LLN doesn't guarantee a specific outcome in any single trial, and it doesn't say how fast the convergence occurs. The speed of convergence depends on the specific problem.

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🧵10/10 - Conclusion 🎉

We've successfully demonstrated the Law of Large Numbers in R using a coin flip example. Remember, this is just a simple example, and you can apply the same logic to more complex problems.

Happy coding! 🚀

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