Expected Value

Definition

Concept

Expected value (EV) or expectation: weighted average of all possible values of a random variable. Represents long-run mean outcome. Denoted E[X] for random variable X.

Mathematical Formulation

EV defined as an integral or summation over all outcomes weighted by their probabilities.

Interpretation

Interpreted as center of mass of distribution, balancing point of probability-weighted outcomes.

Discrete Random Variables

Definition

For discrete X with values x_i with probabilities p_i: E[X] = Σ x_i p_i.

Conditions

Sum of p_i = 1; series Σ |x_i| p_i convergent (finite expectation).

Example

Die roll: values 1 to 6, each with p=1/6; E[X] = (1+2+3+4+5+6)/6 = 3.5.

Table: Discrete Distribution Example

Continuous Random Variables

Definition

For continuous X with probability density function (pdf) f(x): E[X] = ∫ x f(x) dx over support.

Conditions

Integral of |x| f(x) finite; pdf integrates to 1 over domain.

Example

Uniform distribution U(a,b): E[X] = (a + b)/2 as pdf = 1/(b - a).

E[X] = ∫_{-∞}^∞ x f(x) dx

Properties

Existence

EV exists if integral or sum of |x| times probability finite.

Uniqueness

EV unique for given distribution.

Monotonicity

If X ≥ Y almost surely, then E[X] ≥ E[Y].

Boundedness

If X bounded by M, then |E[X]| ≤ M.

Linearity of Expectation

Definition

E[aX + bY] = aE[X] + bE[Y] for any random variables X,Y and constants a,b.

Independence

Independence not required for linearity.

Extension

Extends to finite or countable sums: E[Σ X_i] = Σ E[X_i].

E( aX + bY ) = a E(X) + b E(Y)

Calculation Techniques

Direct Summation/Integration

Apply definition using pmf/pdf.

Using Moment Generating Functions

Derive EV as first derivative of MGF at zero.

Law of the Unconscious Statistician

Calculate E[g(X)] = Σ g(x_i) p_i or ∫ g(x) f(x) dx without pdf of g(X).

Conditional Expectation

E[X] = E[ E[X|Y] ] (Law of total expectation).

Examples

Simple Discrete

Bernoulli(p): E[X] = p.

Binomial(n,p)

E[X] = np.

Geometric(p)

E[X] = 1/p.

Continuous Uniform

E[X] = (a+b)/2.

Normal Distribution

E[X] = μ (mean parameter).

Applications

Decision Theory

Optimize choices by maximizing expected utility or payoff.

Statistics

Expected value as estimator of central tendency and population mean.

Finance

Calculate expected returns, risk assessment, pricing derivatives.

Game Theory

Evaluate strategies based on expected payoffs.

Machine Learning

Loss functions and expected risk minimization.

Relation to Variance

Definition

Variance Var(X) = E[(X − E[X])²] quantifies spread around expectation.

Formula

Var(X) = E[X²] − (E[X])²

Implications

EV alone insufficient to describe variability; variance complements expectation.

Expected Value in Common Distributions

Bernoulli

E[X] = p.

Binomial

E[X] = np.

Poisson

E[X] = λ.

Uniform

E[X] = (a + b)/2.

Normal

E[X] = μ.

Limitations and Misconceptions

Non-existence

EV may not exist if distribution has heavy tails (e.g. Cauchy distribution).

Not a Typical Outcome

EV may be non-attainable or non-representative of typical outcomes.

Misuse in Decision Making

Ignoring variance and risk leads to misleading conclusions.

Extensions and Generalizations

Conditional Expectation

Expectation conditioned on another variable; key in stochastic processes.

Vector-valued Random Variables

EV defined component-wise for random vectors.

General Measure-Theoretic Definition

Expectation as Lebesgue integral with respect to probability measure.

Higher Moments

Expectation extends to moments of all orders: E[Xⁿ].

References

• Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1, Wiley, 1968, pp. 45-80.
• Ross, S. M. A First Course in Probability, 9th ed., Pearson, 2014, pp. 105-140.
• Grimmett, G., Stirzaker, D. Probability and Random Processes, 3rd ed., Oxford University Press, 2001, pp. 60-95.
• Billingsley, P. Probability and Measure, 3rd ed., Wiley, 1995, pp. 130-160.
• Casella, G., Berger, R. L. Statistical Inference, 2nd ed., Duxbury Press, 2002, pp. 200-230.