Joint cumulative distribution function

The joint cumulative distribution function of a multivariate random variable $⃗ 𝐗 = (𝑋 𝑖) 𝑛 𝑖 = 1$ is prob

𝐹 ⃗ 𝐗 (⃗ 𝐱) = ℙ (𝑛 ⋂ 𝑖 = 1 {𝑋 𝑖 \leq 𝑥 𝑖})

In order to find the cumulative distribution function of a single component $𝑋 𝑖$ of $⃗ 𝐗$ , called its marginal cumulative distribution function, one takes the limit of all other components

ℙ (𝑋 𝑖 \leq 𝑥 𝑖) = l i m 𝑥 𝑗 \to \infty, \forall 𝑗 \neq 𝑖 𝐹 ⃗ 𝐗 (⃗ 𝐱)

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Joint cumulative distribution function

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