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Probability, Statistics and Financial Data / Continuous Random Variables

Cumulative Distribution Functions

The PMF and the PDF are described by fundamentally different mathematical objects — a discrete list of probabilities versus a density integrated over ranges — which is awkward whenever a problem needs to compare or combine random variables of both types, or whenever the type is not settled up front. The cumulative distribution function (CDF), FX(x)=P(X≤x)F_X(x)=\mathbb{P}(X\le x), is defined identically for every random variable regardless of type, and this note shows it is the natural unifying object: the PMF and PDF are both recoverable from it (by differencing or differentiating), and it is often the easiest route to a distribution — computed first, then differentiated or differenced to recover the PMF/PDF, as the note's central worked example (the maximum of several random variables) demonstrates.
Learning objectives. By the end of this note you should be able to:
  • define the CDF and state its defining properties (monotonicity, limits, and the discrete/continuous distinction);
  • recover a PMF or PDF from a CDF by differencing or differentiating;
  • derive the CDF of max⁡(X1,…,Xn)\max(X_1,\ldots,X_n) for independent random variables and use it to find the corresponding PMF/PDF;
  • state and use the exponential distribution's memoryless property.

In this lesson

  1. The CDF as a unifying object
  2. Deriving a distribution via its CDF: the maximum of independent random variables
  3. The Exponential's memoryless property, via the CDF

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