Let's look at a simple example to see what the famous Monte Carlo Method does. What's the average height of all people living in the United States? Well, this is impossible to determine exactly because we would have to measure the heights of all people in the United States. But we've already seen that we can actually estimate that average quite well. All we do is simply take a sample of size 100 and then we use the average height of that sample as an estimate of the average height of all the people. Remember, we talk about a parameter like theta when we talk about the population. So in this case, theta would be the average height of the population. We estimate that parameter theta with a statistic which we call theta hat, and that statistic is based on a sample. In this case, our statistic would be simply the average of the sample. We already know that the sample mean tends to be close to the population mean even for moderate sample sizes such as 100, and that's because of the law of large numbers. This is a simple example of the Monte Carlo method. Sometimes it's simply called simulation. What that method does is, it approximates a fixed quantity theta by the average of independent random variables that have expected value equal to theta. Then, by the law of large numbers, the approximation error can be made as small as you wish by using a large enough sample size. It turns out that the Monte Carlo method can also be used for more complicated quantities. One quantity which is important in statistics is the standard error of a statistic. Remember that the standard error tells you roughly how far off the statistic will be from its expected value. There's a formal definition of the standard error which is given there. It's simply the square root of the variance when we think about theta hat as a random variable. So how would the Monte Carlo method work to estimate the standard error in this case? In the first step, we would get many, let's say 1,000 samples of 100 observations each. Remember we take 100 observations each because the standard error of Theta hat is based on 100 observations. Next we compute theta hat for each of these 1,000 samples, and that gives us 1,000 copies of these estimates which we call theta hat 1 up to theta hat 1,000. Finally, we simply compute the standard deviation of these 1,000 estimates. The formula you see there is simply the formula for the standard deviation. The average of these estimates is simply theta hat bar, remember that the bar notation denotes averages. I didn't want to put two superscripts on there to make it not too confusing, so I wrote down the average of the theta hats. So note this is not an average of independent random variables because the average of the theta hats is part of each of these terms, and so, that makes the whole thing dependent, but the Monte Carlo method still works. It turns out that this quantity is roughly equal to the standard error. So while this example looks a bit more complicated than the previous one, what's really going on here is, that it's simply an application of the law of large numbers. But the caveat is that this method will only work if we can draw many samples of size 100. In other words, simulation works if I can sample as much as I wish.