• A random variable is often denoted \(X\)
• We don't know its value a priori, but we can characterize its behavior via its distribution
  • Describes relative frequency with which values occur
  • AUC = 1 = 100%

Its mean \(\mu\) and standard deviation \(\sigma\).

• We need to distinguish the true mean \(\mu\)
• From the sample mean based on a sample \(X_1, \ldots, X_n\): \(\overline{X}_n = \frac{1}{n}\sum_{i=1}^{n}X_i\)
• Moral of the Story: \(\overline{X}_n\) estimates \(\mu\)

For a given \(\epsilon\) and \(n\)

• Let \(X\) have \(\mu=0\)
• Based on a sample \(X_1, \ldots, X_n\) of size \(n\), compute \(\overline{X}_n\)
• Do this a bunch of times to see how \(\overline{X}_n\) varies from sample to sample

Recall \(\mu=0\). For \(\epsilon=0.1\)

Let \(X\) be a RV with mean \(\mu\) and \(\sigma<\infty\). Then

1. The standard deviation of \(\overline{X}_n = \sqrt{\frac{\sigma^2}{n}}\)
2. Chebyshev's Inequality:

• Example: Let \(k=2\), \(\mathbb{P}\left(\left|X-\mu\right| \geq 2\sigma\right) \leq \frac{1}{4}\)
• For ANY RV, the prob that it lies outside 2 standard deviations of its mean is bounded by 25%.

Normal RV with \(\mu=0\) and \(\sigma=1\)

Gamma RV with \(\mu=\) 1.25 and \(\sigma=\) 0.559