4.4 Chapter Summary

This chapter introduced expectations as a tool for summarizing the center and breadth of a distribution.

Interactive Tools:¶

1. Law of Large Numbers Interactive - Use this interactive to watch sample averages over large data sets converge to expectations.

Expectations¶

Definitions and examples are all available in Section 4.1.

1. The expected value of a random variable, $\mathbb{E}[X]$, is the weighted average of possible $x$ against the PMF/PDF:

   $$\mathbb{E}[X] = \begin{cases} \sum_{\text{all } x} x \text{PMF}(x) & \text{ if discrete} \\ \int_{\text{all } x} x \text{PDF}(x) dx & \text{ if continuous} \end{cases}$$

   (1)

   • The expected value is equivalent to the center of mass of the distribution

   • Long run sample averages converge to the expected value

2. The expected value of a function of a random variable, $\mathbb{E}[f(X)]$, is the weighted average over each $x$, of $f(x)$, weighted by the PMF/PDF

1. The expected value is distinct from the:

   • Mode: the most likely outcome, or collection of outcomes that maximize the PMF/PDF.

   • Median: the “midpoint” value $x$ such that $\text{Pr}(X < x_*) = \text{Pr}(X > x_*)$.

Rules of Expectations¶

Definitions and examples are all available in Section 4.2.

1. Expectations of key distributions:

   • Constants: $\mathbb{E}[c] = c$.

   • Indicators: if $X \sim \text{Bernoulli}(p)$, then $\mathbb{E}[X] = p$.

   • Symmetric: If $X$ is drawn symmetrically about $x_*$, then $\mathbb{E}[X] = x_*$.

   • Binomial: If $X \sim \text{Binomial}(n,p)$, then $\mathbb{E}[X] = n p$.

2. Linearity: $\mathbb{E}[a X + b] = a \mathbb{E}[X] + b$.

3. Additivity: for any pair of random variables $X$ and $Y$, $\mathbb{E}[X + Y] = \mathbb{E}[X] + \mathbb{E}[Y]$.

We’ll add more properties to this list in future chapters.

Variance:¶

Definitions and examples are all available in Section 4.3.

1. Given, $\mathbb{E}[X] = \bar{x}$, the variance and standard deviation of a random variable are:

   $$\text{Var}[X] = \mathbb{E}[(X - \bar{x})^2], \quad \text{SD}[X] = \sqrt{\text{Var}[X]}$$

   (3)

   • The standard deviation measures the breadth, spread, or width of the distribution

2. Properties of Variance:

   • $\text{Var}[X] \geq 0$

   • $\text{Var}[c] = 0$

   • $\text{Var}[X + b] = \text{Var}[X]$

   • $\text{Var}[a X] = a^2 \text{Var}[X]$

3. To compute variances, we often use:

   $$\text{Var}[X] = \mathbb{E}[X^2] - \mathbb{E}[X]^2$$

   (4)

   • The variance in a random variable is its expected square, minus its squared expectation.