Distribution of sum of random variables (incomplete)

Dependencies: (incomplete)

  1. Random variable
  2. Group

Let $X$ and $Y$ be 2 random variables with the same support $D$, where $(D, +)$ is a group. Let $Z = X + Y$.

Let $X$ and $Y$ be discrete. Let $f_{X,Y}$ be the joint probability mass function of $X$ and $Y$ and let $f_Z$ be the probability mass function of $Z$. Then \[ f_Z(z) = \sum_{y \in D} f_{X, Y}(z - y, y) = \sum_{x \in D} f_{X, Y}(x, z-x). \]

Let $X$ and $Y$ be continuous random variables and let $+$ (as usual) correspond to addition of real numbers. Let $f_{X,Y}$ be the joint probability density function of $X$ and $Y$ and let $f_Z$ be the probability density function of $Z$. Then \[ f_Z(z) = \int_{-\infty}^{\infty} f_{X, Y}(z - y, y) dy = \int_{-\infty}^{\infty} f_{X, Y}(x, z-x) dx. \]

Proof for discrete random variables

\begin{align} f_Z(z) &= \Pr(X + Y = z) \\ &= \sum_{y \in D} \Pr(X + Y = z \cap Y = y) \\ &= \sum_{y \in D} \Pr(X = z - y \cap Y = y) \\ &= \sum_{y \in D} f_{X, Y}(z - y, y) \end{align}

\begin{align} f_Z(z) &= \Pr(X + Y = z) \\ &= \sum_{x \in D} \Pr(X + Y = z \cap X = x) \\ &= \sum_{x \in D} \Pr(X = x \cap Y = z - x) \\ &= \sum_{x \in D} f_{X, Y}(x, z - x) \end{align}

Proof for continuous random variables

(Proof pending)

Dependency for:

  1. Sum of independent normal random variables is normal

Info:

Transitive dependencies:

  1. /analysis/topological-space
  2. /sets-and-relations/countable-set
  3. /sets-and-relations/de-morgan-laws
  4. σ-algebra
  5. Generated σ-algebra
  6. Borel algebra
  7. Measurable function
  8. Generators of the real Borel algebra (incomplete)
  9. Measure
  10. σ-algebra is closed under countable intersections
  11. Group
  12. Probability
  13. Random variable