The Log-Normal Distribution - Sample Estimates and the CLT
The Log-Normal Distribution - Sample Estimates and the CLT
To explore the central limit theorem (CLT) and its applicability, we begin by generating a large sample of 10,000 random observations from a lognormal distribution with parameters µ = 2 and σ² = 0.25. By using a seed value of 100 to ensure reproducibility, the first few observations from this sample can be displayed using the head function. This step demonstrates how well the simulated data aligns with the theoretical lognormal distribution defined by the given parameters.
Next, we compute the sample mean, median, and variance from the generated sample and compare these statistics with the corresponding values of a population following a lognormal distribution with the same parameters. This comparison helps us understand the accuracy of our sample statistics as estimators of the true population parameters. By examining these values, we assess the degree to which our sample represents the underlying distribution and whether it provides reliable estimates.
Finally, treating the generated data as the population, we draw 500 different random samples, each of size 200, from this population and calculate the sample mean for each sample. Using these means, we plot a histogram to visualize their distribution. Interpreting the histogram allows us to observe the emergence of a normal distribution of sample means, thus illustrating the central limit theorem in practice. This process highlights the power of CLT in approximating the distribution of sample means, regardless of the original population's distribution.
Introduction to R for Actuarial Students - CS1B - Paper D - Question 2