Difficulty: Intermediate | Prerequisites: probability distributions, p.d.f., c.d.f., integration basics, random variables.
Expected value is the centrepiece of probability theory: it tells you where a random variable "lives" on average. This lecture extends that idea into two directions. First, it shows what happens when you apply a non-linear function to a random variable (option payoffs, asset valuations), where Jensen's Inequality becomes the key result. Second, it introduces medians and quantiles as alternative ways to summarise a distribution's location. If you are comfortable computing integrals and sums of p.d.f.s and p.m.f.s, you have the tools you need.
The expected value of a random variable is its weighted-average outcome (sum for discrete, integral for continuous). Expectation is linear, but applying a non-linear function to a random variable before taking the expectation changes things: E[u(X)] is generally not equal to u(E[X]). Jensen's Inequality pins down the direction of that gap for convex functions, and this has direct consequences for valuing options, risky assets, and any payoff with a "kink."
Expected value (expectation), E[X]
A measure of the general location (centre) of a random variable's distribution. For a discrete random variable, it is the probability-weighted sum of all possible values. For a continuous random variable, it is the integral of x times the p.d.f. over the support.
In simple terms, it is the long-run average you would get if you could repeat the random experiment forever.
Linearity of expectation
The property that E[aX + bY] = aE[X] + bE[Y] for any constants a, b and random variables X, Y. This holds regardless of whether X and Y are independent.
Think of it as: you can "pull apart" a weighted combination of random variables and take each expectation separately.
Jensen's Inequality
If u(x) is a convex function, then E[u(X)] ≥ u(E[X]). If u(x) is concave, the inequality flips.
In simple terms, pushing a random variable through a "curving-up" function inflates the average outcome relative to just plugging the average into the function.
Option value
The value of flexibility, i.e. the ability to wait until uncertainty resolves before committing to a decision. In the lecture's example, the payoff u(x) = max{0, x} is the classic call-option payoff.
Think of it as: you only act when the outcome is favourable, so the right to choose is always worth at least zero.
Convex function
A function u(x) where any chord between two points on its graph lies on or above the graph itself. Equivalently, the second derivative (when it exists) is non-negative.
In simple terms, the curve bends upward, like the shape of x² or the call-option payoff max{0, x}.
An option gives you the right, not the obligation, to act after observing an outcome.
The call-option payoff is u(x) = max{0, x}. You exercise only when x > 0.
Because u(x) = max{0, x} is a weakly convex function, Jensen's Inequality applies directly: E[u(X)] ≥ u(E[X]) = u(0) = 0.
The exact value of E[u(X)] depends on the full distribution of X, not just its mean.
Consider an asset whose returns grow at a constant rate g. Its value, discounted at rate r, is V(g) = 1 / (r - g), provided g < r.
Two assets can share the same expected growth rate but produce very different expected valuations, because V(g) is a convex function of g.
Risky stock: E[G₁] = 0.9 × 0% + 0.1 × 10% = 1%. Expected valuation E[V(G₁)] = 0.9 × (1/0.15) + 0.1 × (1/0.05) = 20/3 + 10/6 = 40/6 ≈ 6.67.
Bond: E[G₂] = 1% with certainty. E[V(G₂)] = 1 / (0.15 - 0.01) = 1/0.14 ≈ 7.14.
Same expected growth rate, different expected valuations. The risky stock has higher convexity exposure, which inflates E[V(G)] relative to V(E[G]).
Expectation is linear: E[aX + bY] = aE[X] + bE[Y].
This holds for any random variables X, Y, whether or not they are independent.
Linearity makes expectation easy to compute for sums and weighted averages.
For a convex function u: E[u(X)] ≥ u(E[X]).
For a concave function u: E[u(X)] ≤ u(E[X]).
The gap between the two sides depends on how much curvature u has and how spread out X is.
When u is linear, E[u(X)] = u(E[X]) exactly, which is just linearity of expectation.
E[X] = \sum_x x \cdot P(X = x) \quad \text{(discrete)}E[X] = \int_{-\infty}^{\infty} x \cdot f_X(x) \, dx \quad \text{(continuous)}E[aX + bY] = aE[X] + bE[Y] \quad \text{(linearity)}u \text{ convex} \implies E[u(X)] \geq u(E[X]) \quad \text{(Jensen's Inequality)}V(g) = \int_0^{\infty} e^{(g - r)t} \, dt = \frac{1}{r - g}, \quad g < ru(x) = \max\{0, x\} \quad \text{(call-option payoff)}Option pricing in finance relies on exactly this setup: a call option pays max{0, S - K}, and Jensen's Inequality guarantees the option always has non-negative value. The growth-rate example mirrors how investors compare a risky equity to a safe bond: even when expected growth rates match, the convexity of the valuation function means the risky asset's expected value can differ substantially.
Students often assume E[u(X)] = u(E[X]) for any function u. This is only true when u is linear. For non-linear functions, the two sides are generally different.
Students sometimes think Jensen's Inequality requires X to have a specific distribution. It does not: it applies to any random variable, provided u is convex (or concave).
A common error is forgetting that the option payoff max{0, X} is itself a non-linear function of X, so its expected value cannot be found simply by plugging E[X] into max{0, ·}.
Some students confuse "linearity of expectation" with "independence." Linearity holds whether or not the random variables are independent.
⚠️ Be ready to compute E[u(X)] for a given distribution and a non-linear u, then compare with u(E[X]) to illustrate Jensen's Inequality.
⚠️ Know how to identify whether a function is convex or concave and state which direction Jensen's Inequality goes.
⚠️ The growth-rate example is a classic exam setup: two assets with the same expected growth rate but different risk profiles, leading to different expected valuations.
⚠️ Linearity of expectation is frequently tested as a shortcut for computing expectations of sums.
True or false: E[X²] = (E[X])² for any random variable X.
False. x² is convex, so by Jensen's Inequality E[X²] ≥ (E[X])², with equality only when X is constant.
Fill in the blank: The call-option payoff is u(x) = ____.
max{0, x}
True or false: Linearity of expectation requires independence.
False. It holds for any random variables.
If u is concave, does E[u(X)] sit above or below u(E[X])?
Below: E[u(X)] ≤ u(E[X]).
True or false: Two assets with the same expected growth rate must have the same expected valuation.
False. The valuation function V(g) = 1/(r - g) is convex in g, so risk matters.
Q: Let X be a continuous random variable with p.d.f. f(x) = 2x on [0, 1]. Compute E[X].
A: E[X] = ∫₀¹ x · 2x dx = 2 ∫₀¹ x² dx = 2 · (1/3) = 2/3.
Q: Using the same distribution, compute E[X²] and verify that E[X²] ≥ (E[X])².
A: E[X²] = 2 ∫₀¹ x³ dx = 2 · (1/4) = 1/2. Since (E[X])² = (2/3)² = 4/9 ≈ 0.444, and 1/2 = 0.5 > 0.444, Jensen's Inequality holds.
Q: A new technology has payoff X where X is equally likely to be -2 or +3. What is the expected value of the option payoff u(X) = max{0, X}?
A: E[max{0, X}] = 0.5 · max{0, -2} + 0.5 · max{0, 3} = 0 + 1.5 = 1.5. Note that u(E[X]) = max{0, 0.5} = 0.5, so E[u(X)] > u(E[X]), consistent with Jensen's Inequality.
Q: Explain why E[V(G)] for a risky stock can exceed V(E[G]) even though both assets have the same expected growth rate.
A: V(g) = 1/(r - g) is a convex function of g. By Jensen's Inequality, the expected value of a convex function of a random variable is at least as large as the function evaluated at the expected value. The riskier stock has more spread in G, which amplifies the convexity effect.
Jensen's Inequality reappears whenever you study risk aversion in economics or finance: a risk-averse agent has a concave utility function, so E[u(W)] < u(E[W]), which is why they pay for insurance. The option-value idea connects to real options in corporate finance (the value of waiting to invest). Linearity of expectation is the workhorse behind variance decomposition (Var(X) = E[X²] - (E[X])²), which you will use repeatedly in later lectures on variance and moment-generating functions.
Expected value, expectation, mean, average, E[X], linearity of expectation, Jensen's Inequality, convex function, concave function, option value, call option, max{0, x}, option payoff, growth rate valuation, risk vs. return, non-linear transformation of random variables, ANST 10141T, Analytical Statistics NYU.