A stock's overnight return can be — a value from a continuum, not a countable list. The Discrete Random Variables topic's PMF, which assigns positive probability to individual values, cannot describe this kind of randomness: for a genuinely continuous quantity, every single point has probability zero, yet the variable is clearly not deterministic. The probability density function (PDF) resolves this: it is not a probability itself, but a "probability per unit length" whose integral over a range gives the probability of landing there. This note builds the PDF from scratch, restates expectation and variance as integrals in place of sums, and introduces the two simplest named continuous distributions — Uniform and Exponential — as the continuous counterparts of the discrete toolkit's Bernoulli-family members.
Learning objectives. By the end of this note you should be able to:
- state the defining properties of a PDF and verify a candidate function is legitimate;
- compute event probabilities, expectation, and variance from a PDF, including via the expected value rule;
- state the PDF, mean, and variance of the Uniform and Exponential distributions;
- explain precisely why for any single point under a continuous distribution, without this implying is "impossible."
In this lesson
- The probability density function
- Expectation and variance for continuous random variables
- The Uniform distribution
- The Exponential distribution
Before you start
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