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Simulation and Analysis of Exponential Distribution Using Inverse Transform Method

Simulation and Analysis of Exponential Distribution Using Inverse Transform Method In this exercise, you will simulate samples from an exponential distribution using the inverse transform method, analyze their means, and examine the application of the Central Limit Theorem (CLT). The task begins with generating 10,000 samples from an Exponential distribution with a rate parameter of 3, each of size n = 10, using the inverse transform method in R. The generated values are saved for later analysis, ensuring reproducibility by initializing the random number generator with a seed value. The next step is to plot a histogram of the means of the samples generated in the first exercise, using an appropriate R function to display the histogram on the probability density scale. This visualization helps to understand the distribution of the sample means. Following this, the Probability Density Function (PDF) of the sampling distribution of the sample mean, as predicted by the CLT, is plotted and superimposed on the histogram. This provides a visual comparison between the empirical sample means and the theoretical distribution under the CLT. Finally, the exercise concludes with a discussion on the sampling distribution of the sample mean and the application of the CLT. By comparing the histogram of the sample means and the PDF graph, you will comment on how well the CLT approximates the distribution of the sample mean in this case. This comprehensive analysis aims to deepen your understanding of the exponential distribution, the inverse transform method, and the implications of the Central Limit Theorem in practical scenarios.