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