Expected product of coordinates in the unit disc

Question

Let $(X,Y)$ be a point chosen uniformly from the disc $x^2+y^2\le1$ . What is $\mathbb{E}[XY]$ ?

Short answer

$0$ . The disc and the uniform density are unchanged by the reflection $(x,y)\mapsto(-x,y)$ , so $(X,Y)$ and $(-X,Y)$ have the same law. Hence $\mathbb{E}[XY]=-\mathbb{E}[XY]$ , which forces it to be zero.

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

See the hint and the full worked solution in the question bank (free account).

Open the worked solution

Lessons behind this question

Continuous Random Variables and Probability Density Functions
Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited

Start with a free lesson: Expectation, Variance and Covariance is part of a free topic. Create a free account, no card needed, to read it in full with worked solutions.

Read it free