Expectation, Moments & Shape

A distribution contains more information than its mean. Expectations, dispersion, and higher moments describe different features and supply the algebra used in sampling theory.

1. Expectation is a probability-weighted average

\[\begin{aligned}\mathbb E[Y]&=\sum_j y_jp_j &&\text{(discrete)},\\ \mathbb E[Y]&=\int_{-\infty}^{\infty}y f_Y(y)\,dy &&\text{(density)},\\ \mathbb E[g(Y)]&=\sum_j g(y_j)p_j\\&\quad\text{or }\int_{-\infty}^{\infty}g(y)f_Y(y)\,dy.\end{aligned}\]

These expectations require the relevant integrability conditions; \(\mathbb{E}|Y|<\infty\) ensures a finite mean. The expectation need not be a possible observation. In the crash example\(, \mathbb{E}[M]=.35\) even though no computer crashes .35 times. A repayment of 110 with probability .99 and zero otherwise has expected repayment 108.90; the expected net payoff after lending 100 is 8.90.

Linearity: \(\mathbb{E}[a+\)bX+cY\(]=a+\)bE\([X]+\)cE[Y]. Independence is unnecessary. However\(, \mathbb{E}[g(Y)]\) generally differs from \(g(\mathbb{E}[Y])\). For example\(, \mathbb{E}[Y^{2}]\) differs from \((\mathbb{E}[Y])^{2}\) whenever \(Y\) has positive variance.

2. Variance and standard deviation

\[\begin{aligned}\mu&=\mathbb E[Y],\\ \sigma^2&=\operatorname{Var}(Y)=\mathbb E[(Y-\mu)^2]\\&=\mathbb E[Y^2]-\mu^2,\\ \sigma&=\operatorname{SD}(Y)=\sqrt{\operatorname{Var}(Y)}.\end{aligned}\]

Variance is nonnegative and is zero only when \(Y\) is constant with probability one. It has squared units; standard deviation has the units of \(Y\). Expanding \((Y-\mu )^{2}\) gives \(\mathbb{E}[Y^{2}]-2\mu \mathbb{E}[Y]+\mu ^{2}=\mathbb{E}[Y^{2}]-\mu ^{2}\).

Worked example. For M in Module \(1, \mathbb{E}[M^{2}]=.10+4(.06)+9(.03)+16(.01)=.77\). Therefore \(\operatorname{Var}(M)=.77-.35^{2}=.6475\) and \(\operatorname{SD}(M)\approx .8047\).

Bernoulli. Since \(Y^{2}=Y, \mathbb{E}[Y]=p, \mathbb{E}[Y^{2}]=p\), and \(\operatorname{Var}(Y)=p(1-p)\). Variance is largest at \(p=.5\) and vanishes at \(p=0\) or 1. This identity later supplies the sampling uncertainty of a sample proportion.

3. Affine transformations and units

\[\begin{aligned}\mathbb E[a+bY]&=a+b\mu,\\ \operatorname{Var}(a+bY)&=b^2\sigma^2,\\ \operatorname{SD}(a+bY)&=|b|\sigma.\end{aligned}\]

Derivation. Subtracting the new mean leaves \(a+\)bY\(-(a+b\mu )=b(Y-\mu )\). Squaring and taking expectation produces \(b^{2}\sigma ^{2}\). Taking the nonnegative square root gives \(|b|\sigma\), including when b is negative.

If after-tax income is \(2000+.8X\), the mean becomes \(2000+.8\mu _{X}\) and the standard deviation becomes \(.8\sigma _{X}. A\) fixed transfer changes location but not spread. Converting dollars to cents multiplies means and standard deviations by 100 and variances by 10,000.

4. Moments, skewness, kurtosis, and outliers

\[\begin{aligned}\text{Raw }r\text{th moment: }&\mathbb E[Y^r],\\ \text{Central }r\text{th moment: }&\mathbb E[(Y-\mu)^r],\\ \operatorname{Skewness}(Y)&=\frac{\mathbb E[(Y-\mu)^3]}{\sigma^3},\\ \operatorname{Kurtosis}(Y)&=\frac{\mathbb E[(Y-\mu)^4]}{\sigma^4}.\end{aligned}\]

Standardized moments require \(\sigma >0\) and existence of the relevant moment. Positive skewness commonly signals a long right tail; negative skewness commonly signals a long left tail. Symmetry implies zero skewness when the third moment exists. The converse is not a general theorem: an asymmetric distribution can have zero third central moment.

Normal kurtosis is 3; excess kurtosis subtracts 3. High kurtosis indicates a large standardized fourth moment and sensitivity to extreme observations. It does not rank every tail probability at every cutoff. Two distributions can share a mean and variance while differing greatly in shape and outlier frequency. Positive changes of units preserve skewness and kurtosis; multiplying by a negative number reverses skewness.

\[\begin{aligned}\mathbb E[(Y-\mu)^3]&=\mathbb E[Y^3]-3\mu\mathbb E[Y^2]+2\mu^3,\\ \mathbb E[(Y-\mu)^4]&=\mathbb E[Y^4]-4\mu\mathbb E[Y^3]\\&\quad+6\mu^2\mathbb E[Y^2]-3\mu^4.\end{aligned}\]

These expansions explain why statements about finite fourth moments appear later: applying a finite-variance LLN to \(Y^{2}\) requires \(\operatorname{Var}(Y^{2})<\infty\). The moment restriction belongs to a theorem'\(s\) justification, not merely to a descriptive statistic.

5. Earnings and model adequacy

SW Table 2.4 compares earnings conditional on education and gender. For women with a high-school diploma, the reported mean is 15.49 and median 14.03; for women with a four-year degree, they are 25.42 and 22.44. The comparison includes dispersion and upper percentiles, not just a mean shift. These are historical textbook summaries of 2012 full-time workers, not current estimates or causal education effects.

The stock-return box in §2.4 supplies a second lesson: matching typical variation with a normal distribution can badly understate extreme outcomes. This concerns the distribution of individual observations; a CLT for averages is a separate claim. Time-varying volatility and heavy tails are two distinct ways to reconsider the model.

Check your understanding

Question. If \(\operatorname{Var}(Y)=9\), what is \(\operatorname{SD}(10-2Y)\)? Is \(\mathbb{E}[Y^{2}]=9\)?

Solution. \(\operatorname{SD}(10-2Y)=6. \mathbb{E}[Y^{2}]=9+\mu ^{2}\), which equals 9 only if \(\mu =0\).

Sources: SW §2.2, pp. 65–71; Table 2.4, pp. 79–80; stock-return box, pp. 85–86; Exercises 2.21 and Appendix 2.1. General sign and moment caveats are explicit mathematical clarifications.