Independence of random variables

Two or more random variables may be defined as independent in much the same way as events. Some random variables $𝑋 𝑖 : 𝜉 \to ℝ$ for $𝑖 = 1, \dots, 𝑛$ are independent iff prob

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

for all ${𝑥 𝑖} 𝑛 𝑖 = 1 \subseteq ℝ$ , i.e. the joint CDF is the product of individual CDFs

Discrete random variables

In the case of discrete random variables the above is equivalent to
$ℙ (𝑛 ⋂ 𝑖 = 1 {𝑋 𝑖 = 𝑥 𝑖}) = 𝑛 \prod 𝑖 = 1 ℙ (𝑋 𝑖 = 𝑥 𝑖)$
for all ${𝑥 𝑖} 𝑛 𝑖 = 1 \subseteq ℝ$ .

A common phrase is independent and identically distributed, often abbreviated as i.i.d..

Conditional independence

Random variables ${𝑋 𝑖} 𝑛 𝑖 = 1$ are conditionally independent given a random variable $𝑌$ iff

ℙ (𝑛 ⋂ 𝑖 = 1 {𝑋 𝑖 \leq 𝑥 𝑖 ∣ 𝑌 = 𝑦}) = 𝑛 \prod 𝑖 = 1 ℙ (𝑋 𝑖 \leq 𝑥 𝑖 ∣ 𝑌 = 𝑦)

for all ${𝑥 𝑖} 𝑛 𝑖 = 1 \subseteq ℝ$ and $𝑦 \in s u p p (𝑌)$ .

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Independence of random variables

Conditional independence

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