Sum of independent normal random variables is normal

Dependencies:

  1. Normal distribution
  2. Independence of random variables (incomplete)
  3. Distribution of sum of random variables (incomplete)

Let $X \sim \mathcal{N}(\mu_X, \sigma_X^2)$ and $Y \sim \mathcal{N}(\mu_Y, \sigma_Y^2)$. Let $X$ and $Y$ be independent. Let $Z = X + Y$. Then $Z \sim \mathcal{N}(\mu_X + \mu_Y, \sigma_X^2 + \sigma_Y^2)$.

Also, $aX + bY \sim \mathcal{N}(a\mu_X + b\mu_Y, a^2\sigma_X^2 + b^2\sigma_Y^2)$.

Proof

https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables#Proof_using_convolutions

Let $X' = aX$ and $Y' = bY$. Then $X' \sim \mathcal{N}(a\mu_X, a^2\sigma_X^2)$ and $Y' \sim \mathcal{N}(b\mu_Y, b^2\sigma_Y^2)$. \[ aX + bY = X' + Y' \sim \mathcal{N}(a\mu_X + b\mu_Y, a^2\sigma_X^2 + b^2\sigma_Y^2) \]

Dependency for:

  1. General multivariate normal distribution

Info:

Transitive dependencies:

  1. /analysis/exponential-grows-superpolynomially
  2. /analysis/integration-by-parts
  3. /measure-theory/linearity-of-lebesgue-integral
  4. /measure-theory/lebesgue-integral
  5. /sets-and-relations/de-morgan-laws
  6. /sets-and-relations/countable-set
  7. /analysis/topological-space
  8. Gaussian integral (incomplete)
  9. Group
  10. Ring
  11. Field
  12. Vector Space
  13. σ-algebra
  14. σ-algebra is closed under countable intersections
  15. Measure
  16. Probability
  17. Conditional probability (incomplete)
  18. Independence of events
  19. Independence of composite events
  20. Generated σ-algebra
  21. Measurable function
  22. Borel algebra
  23. Generators of the real Borel algebra (incomplete)
  24. Random variable
  25. Expected value of a random variable
  26. Distribution of sum of random variables (incomplete)
  27. Linearity of expectation
  28. Variance of a random variable
  29. Normal distribution
  30. Independence of random variables (incomplete)