Expectation, Variance & Moments

Expectation answers “where does this random quantity balance on average?” Variance answers “how far can individual outcomes wander from that center?” A strategy can have positive expectation and still lose today, this week, or across a long unlucky sequence.

Formulas and variables #

For discrete outcomes $x_i$ with probabilities $P_i$,

$$E[X]=\sum_i P_i x_i.$$

The variance and standard deviation are

$$Var(X)=E[(X-\mu)^2]=E[X^2]-\mu^2, \qquad \sigma=\sqrt{Var(X)},$$

where $\mu=E[X]$. Squaring prevents positive and negative deviations from cancelling; the square root returns the spread to the original units.

Life example #

A commute takes 20 minutes with probability 0.8 and 60 minutes with probability 0.2. Its expected duration is

$$0.8(20)+0.2(60)=28\text{ minutes}.$$

No commute takes 28 minutes. Expectation is a probability-weighted planning number, not necessarily a possible observation.

Binary contract at 65% #

Buy one YES share at price $p$ when your probability is $q=0.65$. Profit is $1-p$ on YES and $-p$ on NO, so

$$EV=q(1-p)+(1-q)(-p)=q-p.$$

At $p=0.60$, $EV=0.05$. Yet the realized trade is either $+0.40$ or $-0.60$—never $+0.05$. The one-trade variance is

$$Var(X)=q(1-q)=0.65(0.35)=0.2275,$$

because subtracting the purchase price shifts both outcomes but does not change their spread. The standard deviation is about 0.477, much larger than the 0.05 edge.

Application to no-hedge #

The quoted discount may create positive estimated expectation, but the position retains binary variance until resolution. No-hedge sizing must be based on the full settlement loss and portfolio dependence, not merely on the small positive $q-p$.