Using the Conditional Expectation and Variance

What this lecture covers

This is a worked recitation problem applying the law of total variance to a joint density. X and Y are uniformly distributed over a parallelogram, and the problem finds the variance of X + Y. It demonstrates a choice that matters in practice: which variable to condition on, and why picking the one that keeps conditional slices a constant width makes the calculation far easier.

After watching, you should be able to read a joint density defined on a region, choose a sensible conditioning variable based on the region's geometry, derive conditional distributions from geometric slices, and combine the resulting pieces using the law of total variance.

Key ideas

• Why condition on X and not Y: slicing the parallelogram at a fixed x always gives an interval of the same width, so the conditional distribution of Y given X is uniform with a constant-width shift; slicing at a fixed y gives varying widths, which would be harder to work with.
• Conditional distribution from geometry: for x in [0, 1], Y given X = x is uniform between x and x + 1, since conditioning on a uniform joint density preserves uniformity within the conditional region.
• Conditional expectation as a linear shift: E[X + Y | X] = x + E[Y|X] = x + (midpoint of the interval) = 2x + 1/2.
• Conditional variance ignores the constant shift: Var(X + Y | X) = Var(Y|X) = 1/12, using the uniform-variance formula (b - a)^2 / 12.
• Law of total variance combines both pieces: Var(X + Y) = Var(2X + 1/2) + E[1/12] = 4*Var(X) + 1/12.
• Marginalizing the joint density: integrating the joint density over y at each x shows the marginal density of X is uniform on [0, 1], giving Var(X) = 1/12 and a final answer of 5/12.

Before you watch

• Watch the main lecture on iterated expectations for the law of total variance.
• Be comfortable computing marginal densities from a joint density, and know the variance formula for a uniform random variable.

Check your understanding

1. Why does conditioning on X, rather than Y, simplify this problem given the shape of the parallelogram?
2. Why does the constant term inside a conditional expectation not affect the conditional variance?
3. Derive the marginal density of X by marginalizing the joint density over y.
4. Walk through both terms of the law of total variance calculation for Var(X + Y).