Definition:Joint Distribution
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Definition
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $\struct {S, \Sigma'}$ be a measurable space.
Let $X_1, X_2, \ldots, X_d$ be random variables on $\struct {\Omega, \Sigma, \Pr}$ taking values in $\struct {S, \Sigma'}$.
Define a random vector $X : \Omega \to S^d$ by:
- $\map X \omega = \tuple {\map {X_1} \omega, \map {X_2} \omega, \ldots, \map {X_d} \omega}$
for each $\omega \in \Omega$.
Then the joint distribution of $X_1, X_2, \ldots, X_d$ is defined as the probability distribution of $X$.
Sources
- 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $10$: Probability: $10.1$: Basics