A radar's speed measurement is noisy — the true speed and the observed reading are two continuous random variables tied together by a shared experiment, and pricing or estimating anything meaningful requires reasoning about them jointly. This note extends Joint PMFs of Multiple Random Variables and Conditioning and Independence for Discrete Random Variables to the continuous case: the joint PDF as a two-dimensional density, marginal densities recovered by integrating out the other variable, the joint CDF, and conditional densities given an event or given the value of another random variable — including a genuinely famous worked example, Buffon's needle, and a noisy-measurement example that sets up the next lesson's continuous Bayes' rule.
Learning objectives. By the end of this note you should be able to:
- define the joint PDF of two continuous random variables and compute probabilities of two-dimensional events from it;
- recover marginal densities by integrating the joint density over the other variable;
- define and compute the conditional density , and use it to build a joint density sequentially;
- apply the total probability formula for densities to compute an unconditional density from regime-conditional densities.
In this lesson
- The joint PDF
- Conditioning on an event
- Conditioning on another random variable
Before you start
- The Normal Distribution
- Joint PMFs of Multiple Random Variables
- Conditioning and Independence for Discrete Random Variables
The full lesson, with worked examples and exercise solutions, is for members.
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