September 20, 2016

## Random Variable

• 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%

## Further Characterizations

Its mean $$\mu$$ and standard deviation $$\sigma$$.

## Sample Mean

• 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 $$\mu$$

## Convergence in Probability

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

## Convergence in Probability

Recall $$\mu=0$$. For $$\epsilon=0.1$$

## Two Results

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%.

## Chebyshev's Inequality

Normal RV with $$\mu=0$$ and $$\sigma=1$$

## Chebyshev's Inequality

Gamma RV with $$\mu=$$ 1.25 and $$\sigma=$$ 0.559