Cauchy-Schwarz inequality for random variables

Dependencies:

1. Random variable
2. Expected value of a random variable
3. Linearity of expectation
4. Inner product space
5. Cauchy-Schwarz Inequality

$\newcommand{\E}{\operatorname{E}}$ $\newcommand{\Xbar}{\overline{X}}$ $\newcommand{\Ybar}{\overline{Y}}$ Let $X$ and $Y$ be two complex-valued random variables. Then $|\E(X\Ybar)|^2 \le \E(|X|^2)\E(|Y|^2)$.

Proof

Let $V$ be the set of all complex-valued random variables.

Lemma 1: $V$ is a vector space over $\mathbb{C}$.

Proof. $(V, +)$ is an abelian group because

• Addition of random variables gives another random variable.
• Addition of random variables is associative.
• 0 is an additive identity.
• $-X$ is the additive inverse of random variable $X$.
• Addition of random variables is commutative.

$V$ is a vector space because

• $(V, +)$ is an abelian group.
• Scalar multiplication is associative.
• Distributivity holds.
• 1 is a unit scalar for $V$.

□

For two random variables $X$ and $Y$, define the inner-product $\langle X, Y \rangle$ as $\E(X\Ybar)$.

Lemma 2: $(V, \langle \cdot \rangle)$ is an inner-product space.

Proof.

• Conjugate symmetry: \[ \langle X, Y \rangle = \E(X\Ybar) = \overline{\E(Y\Xbar)} = \overline{\langle Y, X \rangle}. \]
• Linearity in first argument (follows from linearity of expectation of random variables): \[ \langle X_1 + X_2, Y \rangle = \E((X_1 + X_2)\Ybar) = \E(X_1\Ybar) + \E(X_2\Ybar) = \langle X_1, Y \rangle + \langle X_2, Y \rangle. \] \[ \langle aX, Y \rangle = \E((aX)\Ybar) = a\E(X\Ybar) = a \langle X, Y \rangle. \]
• Positive definiteness: \[ \langle X, X \rangle = \E(|X|^2) \ge 0. \] \[ \langle X, X \rangle = 0 \iff \E(|X|^2) = 0 \iff X = 0. \tag*{□} \]

On applying the Cauchy-Schwarz inequality for inner-product spaces, we get \[ |\E(X\Ybar)|^2 = |\langle X, Y \rangle|^2 \le \langle X, X \rangle \langle Y, Y \rangle = \E(|X|^2)\E(|Y|^2). \]

Dependency for: None

Info:

• Depth: 8
• Number of transitive dependencies: 25

Transitive dependencies:

1. /measure-theory/linearity-of-lebesgue-integral
2. /measure-theory/lebesgue-integral
3. /complex-numbers/conjugation-is-homomorphic
4. /sets-and-relations/de-morgan-laws
5. /sets-and-relations/countable-set
6. /analysis/topological-space
7. Group
8. Ring
9. Field
10. Vector Space
11. Inner product space
12. Zero in inner product
13. Inner product is anti-linear in second argument
14. Cauchy-Schwarz Inequality
15. σ-algebra
16. σ-algebra is closed under countable intersections
17. Measure
18. Probability
19. Generated σ-algebra
20. Measurable function
21. Borel algebra
22. Generators of the real Borel algebra (incomplete)
23. Random variable
24. Expected value of a random variable
25. Linearity of expectation