Michael Brenndoerfer | Data & AI, Private Equity, Technology

Michael Brenndoerfer

Probability Theory Fundamentals › Expected Value

Michael BrenndoerferOctober 19, 20254 min read

Quantitative FinanceAll chapters

Part 1 of 15Mathematical Foundations

Chapter 2Probability Theory Fundamentals › Expected Value

Reading section 5 of 15 · Expected Value · 4 minSection 5 of 15 · 4 min

Section 5 of 15 · 4 min read

Probability Theory Fundamentals › Expected ValueOne section · Sections in this chapter5 / 15 ↓

Sections in this chapter15

Reading tools Definitions & inline links

Choose your expertise level to adjust how many terms are explained. Beginners see more tooltips, experts see fewer to maintain reading flow. Hover over underlined terms for instant definitions.

Article links

Make inline references clickable

When it exists, expected value is the distribution's probability-weighted mean. A long-run sample average converges to it only under law-of-large-numbers conditions such as iid repetitions with finite expectation, or an appropriate ergodic/dependence condition.

Expected Value

For a discrete random variable with PMF $p_X(x)$ and finite expectation, meaning $\sum_x |x|p_X(x)<\infty$ , the expected value is:

E[X] = \sum_x x \cdot p_X(x)

For an absolutely continuous random variable with PDF $f_X(x)$ and finite expectation, meaning $\int_{-\infty}^{\infty}|x|f_X(x)\,dx<\infty$ :

E[X] = \int_{-\infty}^{\infty} x \cdot f_X(x)\, dx

where:

$E[X]$ : expected value (also called expectation or mean) of random variable $X$
$x$ : possible values that $X$ can take
$p_X(x)$ : probability mass function giving $P(X = x)$ for discrete $X$
$f_X(x)$ : probability density function for continuous $X$
$\sum_x$ : sum over all possible values of $X$
$\int_{-\infty}^{\infty}$ : integral over the entire real line

The expected value is a probability-weighted average. Each possible value contributes according to both its magnitude and its probability (or density), and these contributions are summed (or integrated). A rare outcome with sufficiently large magnitude can therefore have a material effect on the mean.

The parallel structure between the discrete and continuous formulas reflects a deeper unity: in both cases, we're computing a weighted average where the weights come from the probability distribution. The sum becomes an integral when the set of possible values becomes continuous, but the underlying logic remains the same.

Financial InterpretationLink Copied

For iid repetitions with finite expectation, the sample average converges to expected value. That statement concerns repetitions under a stable distribution; it does not imply that one stock's time average must approach a fixed annual expectation when its return process changes over time.

Consider a simplified binary stock model:

In[9]:

Code

import numpy as np

# Binary model: stock either goes up 20% or down 10%
outcomes = np.array([0.20, -0.10])  # Possible returns
probabilities = np.array([0.60, 0.40])  # Probabilities

# Expected return
expected_return = np.sum(outcomes * probabilities)

Out[10]:

Console

Possible outcomes: up 20% with prob 60%, down -10% with prob 40%
Expected return: E[R] = 20% × 60% + (-10%) × 40% = 8.0%

The expected return of 8% is not a possible outcome of one trial. For iid repetitions of this binary experiment, the sample average converges to 8% by the law of large numbers; any individual trial still produces either a 20% gain or a 10% loss.

Properties of Expected ValueLink Copied

Expected value satisfies several useful properties that make it indispensable for financial calculations:

Linearity: For random variables $X$ and $Y$ with finite expectations and constants $a, b$ :

E[aX + bY] = aE[X] + bE[Y]

where:

$a, b$ : constants (real numbers)
$X, Y$ : random variables (can be dependent or independent)
$E[\cdot]$ : expected value operator
$aX + bY$ : linear combination of the random variables
$aE[X] + bE[Y]$ : the same linear combination applied to the expected values

This property holds even if $X$ and $Y$ are dependent. The expectation "passes through" the linear combination regardless of any relationship between the variables. This is not true for most other summary measures. The variance of a sum, for example, depends critically on the correlation between variables. Linearity makes portfolio expected return calculations straightforward: the expected return of a portfolio is the weighted average of individual expected returns. If you hold 60% stocks with expected return 10% and 40% bonds with expected return 4%, the portfolio expected return is simply $0.6 \times 10\% + 0.4 \times 4\% = 7.6\%$ , regardless of how stocks and bonds move together.

Expectation of a function: For a measurable function $g$ whose expectation is finite, we can use the PMF of a discrete $X$ or the PDF of an absolutely continuous $X$ :

E[g(X)] = \sum_x g(x) \cdot p_X(x) \quad \text{(discrete)}

E[g(X)] = \int_{-\infty}^{\infty} g(x) \cdot f_X(x)\, dx \quad \text{(continuous)}

where:

$g(X)$ : any function applied to the random variable $X$
$g(x)$ : the function evaluated at a specific value $x$
$p_X(x)$ : probability mass function (for discrete $X$ )
$f_X(x)$ : probability density function (for continuous $X$ )

This result, known as the Law of the Unconscious Statistician (LOTUS), is useful when the stated expectation is finite: we weight $g(x)$ by the original distribution of $X$ without first deriving the distribution of $g(X)$ . It supports moment calculations and model-based derivative payoffs subject to their integrability conditions.