Random variables and expectation

A random variable is a function from the sample space Ω to the real numbers. Intuitively, it is a numerical quantity whose value depends on a random outcome. Stock prices, daily returns, credit losses, recovery rates — every quantitative object in finance is a random variable.

Discrete vs continuous

A discrete random variable takes countably many values (e.g., number of defaults in a portfolio). Its distribution is the PMF p(x) = P(X = x). A continuous random variable takes uncountably many values (e.g., a stock price). Its distribution is the PDF f(x), and P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx. Both have a CDF F(x) = P(X ≤ x), which is the most universally usable object.

Expectation and variance

E[X] = Σ x·p(x)     (discrete)
E[X] = ∫ x·f(x) dx  (continuous)
Var(X) = E[(X - E[X])²] = E[X²] - (E[X])²

Variance of a linear combination

Var(aX + bY) = a² Var(X) + b² Var(Y) + 2ab Cov(X, Y)

Unlike expectation, variance is not linear; the covariance term is the diversification lever that drives all of portfolio theory. In matrix form for w ∈ Rⁿ: Var(wᵀr) = wᵀΣw — the quadratic form we'll see in Portfolio Theory.

Moments and the moment-generating function

The k-th raw moment is E[X^k]; the k-th central moment is E[(X - E[X])^k]. The MGF M(t) = E[e^(tX)] generates all moments by differentiation. Two distributions with identical MGFs (in a neighbourhood of 0) are identical.

• Mean (1st moment): central tendency.
• Variance (2nd central moment): spread.
• Skewness (3rd standardised moment): asymmetry. Negative skew in equity returns means crash risk.
• Kurtosis (4th standardised moment): tail weight. Normal has kurtosis 3; daily equity returns are usually 5-15 (fat-tailed).

Log returns vs simple returns

Simple return: R_t = P_t/P_{t-1} - 1. Log return: r_t = log(P_t/P_{t-1}) = log(1 + R_t). Two reasons quants prefer log returns:

• Time-additive: log return over k periods is the sum of single-period log returns. The CLT then gives approximate normality for moderate-horizon returns.
• Symmetric around zero: a +50% then -50% gives log returns +0.405 and -0.693, but simple returns +50% and -50% give a net -25%. Log returns honour the asymmetric multiplicative reality.