M.3.3 Distributions & Expectation

Basic Prerequisites: 02 · Probability Spaces & Filtrations.

---

1. Intuition & Practical Objective

A random variable is a function on outcomes, and its distribution is the probability it puts on each value range - two random variables can share a law while being different functions, and one variable can have two laws under two measures (Shreve II Def 1.2.3). "$\mathbb{E}[h(X)]$" is just the probability-weighted average of $h$ over the law of $X$, computed by the standard machine (indicator $\to$ simple $\to$ nonnegative $\to$ general) or, when a density exists, as $\int h(x)f_X(x)dx$.

The practical objective: understand the three linked faces of a distribution - (1) the law ${\mu }_X(B)=\mathbb{P}\{X\in B\}$ and its density; (2) expectation as an integral against the law, computable by inverse transform for simulation; (3) independence and the crucial fact that uncorrelated $\nRightarrow$ independent outside joint normality. This is the vocabulary every simulation (Glasserman Ch 2) and every moment/risk computation uses.

---

2. Mathematical Ground Truth & Derivations

2.1 Random variables, laws, densities (Shreve II Def 1.2.3; Shreve I Ch 11)

$X:\Omega \to \mathbb{R}$ is a random variable if $X^{-1}(B)\in \mathcal{F}$ for every Borel $B$. Its law (distribution measure) is

$${\mu }_X(B)=\mathbb{P}\{X\in B\}=\mathbb{P}\{X^{-1}(B)\}.$$

$X$ has a density $f_X:\mathbb{R}\to [0,\infty )$ iff ${\mu }_X(B)={\int }_B{f}_X(x)dx$, i.e. iff ${\mu }_X\ll \text{Leb}$ (absolutely continuous w.r.t. Lebesgue); $f_X$ is the Radon–Nikodym derivative $d{\mu }_X/dx$ (Shreve I Ch 11). Two random variables sharing a law are different objects - “the RV and its distribution are distinct” (Shreve II Def 1.2.3).

2.2 Expectation via the standard machine (Shreve II §1.4; Shreve I Thm 3.32)

$$\mathbb{E}[h(X)]={\int }_{\Omega }h(X)d\mathbb{P}={\int }_{\mathbb{R}}h(x)d{\mu }_X(x)={\int }_{\mathbb{R}}h(x)f_X(x)dx.$$

The construction is inductive: indicator $\to$ simple $\to$ nonnegative $\to$ general (monotone convergence for the nonnegative step), and the limit theorems (MCT, Fatou, DCT) make passing limits through expectations legal.

2.3 Inverse-transform sampling (Glasserman §2.2, eq. 2.13–2.14)

$$X=F^{-1}(U),F^{-1}(u)=\inf \{x:F(x)\ge u\},U\sim \text{Unif}[0,1].$$

This needs exactly one uniform per sample (minimal dimension - key for QMC) and gives $X=-\theta \log (1-U)\sim \text{Exp}(\theta )$ for the exponential, ${\sin }^2(U\pi /2)$ for the arcsine, etc. (Glasserman eqs. 2.13–2.14, p-070).

2.4 Independence: sets, $\sigma$-algebras, variables (Shreve II §2.2)

Independent sets: $\mathbb{P}(A\cap B)=\mathbb{P}(A)\mathbb{P}(B)$; random variables independent if $\sigma (X),\sigma (Y)$ are. Criterion (Shreve II Thm 2.2.7(vi)): joint cdf / joint MGF factors. Uncorrelated $\nRightarrow$ independent: with $X$ standard normal and $Z=\pm 1$ equally, $Y=ZX$ is standard normal, $\mathbb{E}[XY]=\mathbb{E}[X^2]\mathbb{E}[Z]=0$ (uncorrelated), yet $X,Y$ are dependent and have no joint density (Shreve II §2.2 Example). Only jointly normal variables make zero covariance imply independence (components independent iff covariance matrix diagonal, Shreve I §11.8).

---

3. Computational Implementation - inverse transform, expectation, and the independence trap

Simulate an exponential by inverse transform and check its mean and survival function; then exhibit the classic uncorrelated-but-dependent pair $Y=ZX$. Stdlib only.

---

4. Failure Modes & First-Principles Breakdowns

1. Confusing the random variable with its distribution. Two RVs can share a law (e.g. $X$ and $-X$ for symmetric $X$) yet be different functions; one RV can have two laws under two measures. “Is $X$ normal?” is a statement about its law, never about the underlying $\omega$ map (Shreve II Def 1.2.3).
2. Uncorrelated $\Rightarrow$ independent (the Gaussian-only shortcut). It holds for jointly normal variables and fails almost everywhere else - the $Y=ZX$ example has zero covariance yet no joint density and clear dependence (Shreve II §2.2). This silently breaks factor/portfolio independence assumptions.
3. Sampling a distribution without a valid generator. Acceptance–rejection consumes an unbounded number of uniforms per sample, is not monotone in its inputs, and is incompatible with quasi-Monte Carlo; inverse transform is one-uniform-per-sample and monotone (Glasserman §2.2). Using an $O(d^3)$-unstable sampler quietly invalidates the downstream estimator.
4. Dividing by $\mathbb{P}(B)$ when $\mathbb{P}(B)=0$. The elementary conditional formula collapses on zero-probability conditioning; that is the road into the measure-theoretic conditional expectation (04 · Conditional Expectation).

---

5. References

• Shreve, Stochastic Calculus for Finance II
• Shreve, Stochastic Calculus for Finance I
• Glasserman, Monte Carlo Methods in Financial Engineering
• Casella & Berger, Statistical Inference

---

6. Connected Graph Bridges

• Back: 02 · Probability Spaces & Filtrations · Index Hub
• Forward: 04 · Conditional Expectation · 05 · Martingales
• Applications: VaR & Expected Shortfall (tail distributions) · Stochastic Calculus (normal increments, lognormal prices)