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Probability · Lecture 32 of 76 · 22:58

Recitation: Uniform Probabilities on a Triangle

Uniform Probabilities on a Triangle on YouTube

Study guide

What this lecture covers

This is a recitation problem, not a lecture, that gives practice manipulating joint, marginal, and conditional densities for two continuous random variables. It picks a concrete case, X and Y uniformly distributed on the triangle with vertices (0,0), (1,0), and (0,1), and works through the full sequence of calculations you would need for any joint density.

It follows the lecture material on multiple continuous random variables and is meant to reinforce it with hands-on computation, including a symmetry trick for finishing off a calculation without extra integrals.

Key ideas

  • Uniform joint density on a region: for a uniform distribution, the density is a constant equal to 1 divided by the area of the region, and 0 outside it.
  • Marginal by integrating out a variable: the marginal density of Y comes from integrating the joint density over all valid x for each y, keeping careful track of the limits of integration.
  • Conditional PDF as joint over marginal: dividing the joint density by the marginal of Y gives the conditional density of X given Y, which here turns out to be uniform on a shrinking interval.
  • Conditional expectation: the expected value of X given Y = y is computed the same way as an ordinary expectation, but weighted by the conditional density.
  • Total expectation theorem: integrating a conditional expectation against the marginal density recovers the unconditional expectation.
  • Symmetry shortcut: when a joint density is unchanged by swapping X and Y, their expectations must be equal, which can close out an equation without further integration.

Walkthrough

Joint PDF and marginal of Y (0:00)

Because X and Y are uniform on the triangle bounded by x + y <= 1, x >= 0, y >= 0, the joint density is a constant equal to 2 inside the triangle (the reciprocal of its area, 1/2) and 0 elsewhere. Integrating this joint density over x from 0 to 1 - y gives the marginal density of Y as 2(1 - y) for y between 0 and 1, a triangular shape that is higher for small y because more x values pair with small y.

Conditional PDF of X given Y (7:10)

Dividing the joint density by the marginal of Y gives a conditional density of 1/(1-y) for x between 0 and 1 - y, which does not depend on x at all. That means, for a fixed y, X is uniformly distributed on [0, 1-y] — a direct consequence of slicing a uniform (flat) joint density, which always leaves a uniform slice.

Conditional and unconditional expectations (12:19)

Since X given Y = y is uniform on [0, 1-y], its conditional expectation is the midpoint, (1-y)/2. Applying the total expectation theorem and splitting the resulting integral expresses the unconditional expectation of X as 1/2 minus half the expectation of Y.

Finishing with symmetry (19:33)

Because the triangle and the joint density are symmetric under swapping x and y, E[X] and E[Y] must be equal. Substituting this into the equation from the total expectation theorem gives E[X] = E[Y] = 1/3.

Before you watch

  • Know how to find a joint density for a uniform distribution over a region and how to integrate it to get a marginal density.
  • Review the definition of conditional density as joint density divided by marginal density.
  • Recall the total expectation theorem from the discrete case, since it is applied here with an integral in place of a sum.

Check your understanding

  1. Why is the joint density in this problem equal to 2 rather than 1?
  2. Why does slicing a uniform joint density always give a uniform conditional density?
  3. How does the total expectation theorem turn a conditional expectation into an unconditional one?
  4. Why does the symmetry between X and Y let you conclude E[X] = E[Y] without further calculation?

Vocabulary

vertex (noun)
A corner point of a shape, such as a triangle.
The triangle has vertices at (0,0), (1,0), and (0,1).
uniform joint density (phrase)
A density that is constant everywhere inside a region and zero outside it.
The uniform joint density on the triangle equals 2 inside it.
limits of integration (phrase)
The starting and ending values used when calculating an integral.
We must track the limits of integration carefully for this triangle.
shrinking interval (phrase)
A range of values that gets smaller depending on another variable.
The conditional density is uniform on a shrinking interval as y grows.
conditional expectation (phrase)
The average value of a variable, calculated given that another variable has a specific value.
The conditional expectation of X given Y is the midpoint of its range.
midpoint (noun)
The value exactly in the middle of a range.
For a uniform variable, the expectation is the midpoint of its interval.
total expectation theorem (phrase)
A rule for finding an overall average by combining conditional averages weighted by their probabilities.
The total expectation theorem turns the conditional expectation into the unconditional one.
symmetry shortcut (phrase)
A trick that uses a balanced pattern to skip extra calculation.
The symmetry shortcut shows E[X] equals E[Y] without more integrals.
swap (verb)
To exchange the positions of two things.
The density stays the same if we swap X and Y.
joint density (phrase)
A function describing how likely combinations of two or more variables are together.
The joint density is constant everywhere inside the triangle.
marginal density (phrase)
The density of one variable alone, found by integrating out the other variable.
The marginal density of Y is 2(1-y) on [0,1].
conditional density (phrase)
The density of one variable given a fixed value of another.
The conditional density of X given Y is uniform on [0, 1-y].
region (noun)
A defined area within a larger space.
X and Y are uniform over the triangular region.
reciprocal (noun)
The value obtained by dividing 1 by a number.
The constant density is the reciprocal of the triangle's area.
triangular (adjective)
Shaped like a triangle.
The marginal density of Y has a triangular shape.
hands-on (adjective)
Involving direct practice rather than just theory.
This recitation gives hands-on computation practice.
manipulate (verb)
To work with and change something skillfully.
The problem gives practice manipulating joint and conditional densities.
integrate out (phrasal verb)
To remove a variable from a joint density by integrating over all its possible values.
We integrate out X to get the marginal density of Y.
slice (noun)
A thin section cut from a larger shape, here for a fixed value of one variable.
A vertical slice of the uniform joint density is itself uniform.
unconditional expectation (phrase)
The overall average value of a variable, not assuming anything about another variable.
The unconditional expectation of X turns out to equal 1/3.
recitation problem (phrase)
An example problem worked in a practice class to apply lecture ideas.
This is a recitation problem practicing joint and conditional densities.
reinforce (verb)
To strengthen understanding of something already learned.
The exercise reinforces the lecture material with hands-on computation.
keep track of (phrase)
To pay careful attention to something so it is not lost or forgotten.
You must keep track of the limits of integration for this triangle.
consequence (noun)
A result that follows from something else.
A uniform conditional density is a direct consequence of a flat joint density.
close out (phrasal verb)
To finish or complete something.
Symmetry can close out an equation without more integration.

Chapters

From the YouTube description

MIT 6.041SC Probabilistic Systems Analysis and Applied Probability, Fall 2013
View the complete course: http://ocw.mit.edu/6-041SCF13
Instructor: Jimmy Li

License: Creative Commons BY-NC-SA
More information at http://ocw.mit.edu/terms
More courses at http://ocw.mit.edu

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