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