Expectation, Variance, and Moment Generating Functions

If probability measures assign weights to outcomes, expectation tells you the weighted average. Variance tells you how spread out the weights are. Together with higher moments and moment generating functions, they form the quantitative toolkit for summarizing and comparing distributions.

In finance, these are not abstract: pricing is expectation (the risk-neutral price of a derivative is $C_0=e^{-rT}{E}^Q[C_T]$), risk is variance (or standard deviation), and the Sharpe ratio is just mean excess return divided by standard deviation. Moment generating functions provide a clean algebraic route to computing moments and proving distributional results (they are the engine behind the CLT proof in lln-and-clt).

This article requires a solid grasp of distributions from random-variables-and-distributions and the measure-theoretic foundation from probability-measures-and-axioms, since expectation in its full generality is defined as a Lebesgue integral.

Key Topics

• Expectation as Lebesgue integral: $E[X]={\int }_{\Omega }X(\omega )dP(\omega )$, why this is more general than Riemann integration
• Linearity of expectation: $E[aX+bY]=aE[X]+bE[Y]$ (no independence required)
• Variance and standard deviation: $\text{Var}(X)=E[(X-\mu {)}^2]=E[X^2]-(E[X]{)}^2$
• Covariance: $\text{Cov}(X,Y)=E[XY]-E[X]E[Y]$, its role in portfolio variance
• Skewness and kurtosis: ${\gamma }_1=E\left[{\left(\frac{X-\mu }{\sigma }\right)}^3\right]$, $\kappa =E\left[{\left(\frac{X-\mu }{\sigma }\right)}^4\right]$. Fat tails ($\kappa >3$) are why normal models underestimate extreme events
• Moment generating functions: $M_X(t)=E[e^{tX}]$, recovering moments via $E[X^n]=M_X^{(n)}(0)$
• Characteristic functions: ${\phi }_X(t)=E[e^{itX}]$ --- always exists (unlike MGFs), uniquely determines the distribution
• Uniqueness theorem: if two distributions have the same MGF (or characteristic function) in a neighborhood of zero, they are identical

Finance Connections

• Derivative pricing = computing an expectation: $C_0=e^{-rT}{E}^Q[\max (S_T-K,0)]$
• Portfolio variance = ${\sum }_{i,j}{w}_i{w}_j\text{Cov}(R_i,R_j)$, which is why covariance drives diversification
• Sharpe ratio = $\frac{E[R]-r_f}{\sigma (R)}$, the canonical risk-adjusted performance measure
• Fat tails: equity returns have excess kurtosis ($\kappa \approx 5\text{–}10$ for daily returns). Models that assume $\kappa =3$ (normal) systematically underestimate tail risk
• MGFs in Black-Scholes: the log-normal MGF connects directly to the pricing formula

Prerequisites

• random-variables-and-distributions

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This article is a roadmap --- content to be developed in future sessions.