Expectation of product of independent random variables (incomplete)

Dependencies: (incomplete)

  1. Random variable
  2. Expected value of a random variable
  3. Independence of random variables (incomplete)

Let $X_1, X_2, \ldots, X_n$ be independent real-valued random variables. Then $\newcommand{\E}{\operatorname{E}}$ \[ \E\left( \prod_{i=1}^n X_i \right) = \prod_{i=1}^n \E(X_i) \]

Proof

For $n=0$ and $n=1$, this is trivially true.

The theorem can also be proven for $n=2$ (proof pending). We can maybe prove this using Fubini's theorem in the general case. For discrete random variables, the proof isn't hard.

For $n \ge 3$, we can prove the theorem by induction. Suppose the theorem is true for up to $n-1$ random variables. If all $X_i$ are independent, then $X_n$ and $X_1X_2\ldots X_{n-1}$ are independent. \begin{align} & \E\left(\prod_{i=1}^n X_i\right) \\ &= \E\left(\prod_{i=1}^{n-1} X_i\right)\E(X_n) \tag{induction hypothesis for 2 variables} \\ &= \left(\prod_{i=1}^{n-1} \E(X_i)\right) \E(X_n) \tag{induction hypothesis for $n-1$ variables} \\ &= \prod_{i=1}^n \E(X_i) \end{align}

Dependency for:

  1. Chernoff bound
  2. Variance of sum of independent random variables

Info:

Transitive dependencies:

  1. /measure-theory/lebesgue-integral
  2. /sets-and-relations/de-morgan-laws
  3. /sets-and-relations/countable-set
  4. /analysis/topological-space
  5. Group
  6. Ring
  7. Field
  8. Vector Space
  9. σ-algebra
  10. σ-algebra is closed under countable intersections
  11. Measure
  12. Probability
  13. Conditional probability (incomplete)
  14. Independence of events
  15. Independence of composite events
  16. Generated σ-algebra
  17. Measurable function
  18. Borel algebra
  19. Generators of the real Borel algebra (incomplete)
  20. Random variable
  21. Expected value of a random variable
  22. Independence of random variables (incomplete)