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Mathematics and Statistics for Financial Risk Management by Michael B. Miller

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CHAPTER 5

Hypothesis Testing & Confidence Intervals

In this chapter we will explore two closely related topics, confidence intervals and hypothesis testing. At the end of the chapter, we will explore applications, including value at risk (VaR).

THE SAMPLE MEAN REVISITED

Imagine we take the output from a standard random number generator on a computer, and multiply it by 100. The resulting data generating process (DGP) is a uniform random variable, which ranges between 0 and 100, with a mean of 50. If we generate 20 draws from this DGP and calculate the sample mean of those 20 draws, it is unlikely that the sample mean will be exactly 50. The sample mean might round to 50, say 50.03906724, but exactly 50 is next to impossible. In fact, given that we have only 20 data points, the sample mean might not even be close to the true mean.

The sample mean is actually a random variable itself. If we continue to repeat the experiment—generating 20 data points and calculating the sample mean each time—the calculated sample mean will be different every time. As we proved in Chapter 3, even though we never get exactly 50, the expected value of each sample mean is in fact 50. It might sound strange to say it, but the mean of our sample mean is the true mean of the distribution. Using our standard notation:

(5.1)

Instead of 20 data points, what if we generate 1,000 data points? With 1,000 data ...

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