Probability via Integration: PDFs, CDFs, and Expected Value

Asset returns, waiting times, volatility estimates, and default probabilities are all continuous random variables: they take values in an interval rather than a finite set. The probability that such a variable $X$ falls in $[a, b]$ is the integral $\mathbb{P}(a \le X \le b) = \int_a^b f_X(x)\,dx$, where $f_X$ is the probability density function (PDF). The Fundamental Theorem of Calculus connects the PDF to the cumulative distribution function (CDF): $F_X(x) = \int_{-\infty}^x f_X(t)\,dt$, so $F_X'(x) = f_X(x)$. Expected value and variance follow as weighted-average integrals. The normal distribution — the most important distribution in finance — is the first example where the density integrates to 1 via an improper integral that cannot be evaluated in closed form using the tools of this course.

Learning objectives. By the end of this note you should be able to: