Measure, Integration & Real Analysis · Problem 12.11

Probability Measures — Sheldon Axler

Mathematics — Measure Theory · variance-of-sum-not-independent

Problem

Give an example of a probability space

(\Omega, \mathcal{F}, P)

and

X, Y \in \mathcal{L}^2(P)

such that

\sigma^2(X + Y) = \sigma^2(X) + \sigma^2(Y)

but

X

and

Y

are not independent random variables.