Distribution of sum of random variables (incomplete)
Dependencies: (incomplete)
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)
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- /analysis/topological-space
- /sets-and-relations/countable-set
- /sets-and-relations/de-morgan-laws
- σ-algebra
- Generated σ-algebra
- Borel algebra
- Measurable function
- Generators of the real Borel algebra (incomplete)
- Measure
- σ-algebra is closed under countable intersections
- Group
- Probability
- Random variable