Central Limit Theorem: How does sample size affect the sampling distribution?

[FactChecker]

@Dale where can I do a Monte Carlo simulation?

I should tell you that there are computer languages and systems that are designed specifically to do Monte Carlo simulations. If you are going to do large simulations, you should look into those.

I would also say that simple problems with analytical solutions are very easy to modify so that the analytical solutions become a nightmare. In those cases, Monte Carlo estimates are often much easier to get and to be confident of. Even if analytical solutions are still possible, the Monti Carlo estimate can provide a good "sanity check" for the analysis.
For instance, suppose in the coin toss example we added the requirement that a Head will not count if 3 of the prior 5 tosses were Heads. That would be trivial to add to the Monte Carlo simulation, but the analytical solution would be more difficult. Although this example seems artificial, the real world often gets complicated like that.

 

[statdad]

Should I have written $\displaystyle \lim_{n \to \infty} \frac{1}{n} \sum_{i = 1} ^n \overline x_i = \mu$? In words, the mean of the sample means approaches the true mean of the population as the number of samples approaches infinity. Not sure if that's the actual statement or not. How would you write down the correct expression? @Dale

What about my question regarding the sample size? Why do we assume the population is infinite?

we say that the sample mean converges in probability to the population mean: that is, given n epsilon, it is true that the limit as n goes to infinity of P(|Xbar - mu| > epsilon) = 0 (or, equivalently, the limit of
P(|Xbar - mu| <= epsilon) = 1).


It's only when the sequence is standardized as done in other posts that the normal distribution comes into play.