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Lesson 1 of 12 · 13 min

Random variables: PMF, PDF and CDF

A random variable is fully described by its distribution: a PMF or PDF shows where the probability sits, and the CDF adds it up from the left.

In short

  • A random variable attaches a number to every outcome of an uncertain event; finance also calls it a stochastic variable.
  • Discrete variables have countable outcomes (the number of defaults); continuous ones can take any value in a range (a return).
  • The PMF gives P(X=x)P(X = x) for a discrete variable. The PDF gives the density of a continuous variable, and probability is the area under it.
  • The CDF F(x)=P(X≤x)F(x) = P(X \le x) works for both kinds, never decreases, and runs from 0 on the far left to 1 on the far right.
  • For a continuous variable the PDF is the slope of the CDF, and P(a≤X≤b)=F(b)−F(a)P(a \le X \le b) = F(b) - F(a).

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Random variables: PMF, PDF and CDF · Statistical Distributions for Financial Asset Prices and Returns