Lecture 9: Multiple Continuous Random Variables

What this lecture covers

The lecture moves from a single continuous random variable to pairs of them. It asks how to describe two random variables that vary together on a continuous scale, and how the ideas already built for discrete random variables (joint distributions, independence, conditioning) carry over once sums become integrals and probability mass functions become densities.

It sits early in the unit on continuous random variables, right after the single-variable density function was introduced. By the end, you can compute a joint density, check independence, find a marginal density by integrating out a variable, and reason about conditional densities as slices of a joint density.

Key ideas

• Joint density function: f(x,y) is a surface over the plane whose volume above any region gives the probability of landing in that region; the total volume under it equals 1.
• Marginal density: the density of one variable alone, found by integrating the joint density over all values of the other variable.
• Independence: X and Y are independent exactly when the joint density factors as the product of the two marginal densities, for every x and y.
• Conditional density: the density of X given Y near a value y; it is the joint density divided by the marginal density of Y, and geometrically it is a rescaled slice of the joint surface.
• Buffon's needle: a needle dropped at random across parallel lines gives a probability of intersection that can be worked out from a joint uniform density on position and angle, and the result connects to Monte Carlo estimation of integrals.
• Multiplication rule for densities: the joint density equals a marginal density times the matching conditional density, mirroring the discrete case.

Walkthrough

Recap of the single-variable density (2:04)

The lecture opens by reviewing the probability density function of one random variable: probabilities come from integrating the density over an interval, the density is non-negative, and it integrates to 1 over the whole line. Expected values of functions of a random variable use the same weighted-average logic as the discrete case, with the density replacing the PMF and an integral replacing a sum.

Defining the joint density (7:07)

Two random variables are jointly continuous if probabilities of events in the (x,y) plane come from integrating a joint density f(x,y) over that region, which is equivalent to computing the volume under a surface. The density must be non-negative and integrate to 1 over the whole plane. Locally, the probability of landing in a small rectangle of side delta is approximately the density at that point times delta squared, which is why the joint density is described as probability per unit area.

Marginal densities and independence (13:12)

To recover the distribution of X alone from a joint density, you integrate the joint density over all y. This mirrors the discrete formula with sums replaced by integrals. Independence is then defined as the joint density factoring into the product of the two marginals for all x and y; intuitively, learning the value of one variable tells you nothing new about the other, and probabilities of the form "X in A and Y in B" can be computed by multiplying separate probabilities.

Buffon's needle (18:20)

A needle shorter than the spacing d between parallel lines is dropped at random, described by the distance from its center to the nearest line and the angle it makes with the lines. Modeling both as independent uniform random variables gives a joint density that is a constant. The event "the needle crosses a line" becomes a region in this two-dimensional space, and integrating the joint density over that region gives the probability 2l/(pi*d). The lecture notes this can be turned around to estimate pi experimentally, and generalizes to the Monte Carlo method for evaluating integrals that are too hard to compute directly, especially in high dimensions.

Conditional densities (31:32)

Conditioning on a continuous variable taking an exact value has probability zero, so the conditional density of X given Y near y is defined as a limit: it equals the joint density divided by the marginal density of Y. Visually, fixing y and looking at how the joint density varies with x gives a slice of the joint surface; dividing by the marginal density rescales that slice so it integrates to 1. When the variables are independent, this conditional density reduces to the plain marginal density of X, since conditioning does not change beliefs about an independent variable.

Breaking a stick twice (40:49)

A stick of length l is broken at a uniformly random point X, then the remaining piece is broken again at a point Y uniform between 0 and X. The lecture builds the joint density from the marginal of X and the conditional of Y given X, derives the marginal density of Y by integrating out X, and computes the expected length of the final piece as l/4, matching the intuitive halving argument for each break.

Before you watch

• Be comfortable with the density function and cumulative distribution function of a single continuous random variable, covered in the previous lecture.
• Review the discrete joint PMF, marginal PMF, independence, and conditional PMF, since this lecture repeats those definitions with integrals in place of sums.
• A basic calculus refresher on double integrals and integrating over regions bounded by lines will help with the worked examples.

Check your understanding

1. Why must a joint density function integrate to 1 over the entire plane rather than over any smaller region?
2. How do you obtain the marginal density of X from a joint density f(x,y)?
3. What condition on the joint density defines independence of two continuous random variables?
4. In Buffon's needle, why are the center's distance to the nearest line and the needle's angle modeled as independent uniform random variables?
5. Why is the conditional density of X given Y defined using an approximate, rather than exact, conditioning event?