Definition: independent and identically distributed
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independent and identically distributed
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Metadata: ID: D200 | shortcut: iid | author: JoramSoch | date: 2024-08-08, 11:37.
Definition: Let $X_i$ for $i = 1,\ldots,n$ be random variables. Then, $X_1, \ldots, X_n$ are called independent and identically distributed (i.i.d.), if (i) they are statistically independent and (ii) they follow the same probability distribution $\mathcal{D}$ with the same parameters $\theta$:
\[\label{eq:iid} X_i \overset{\mathrm{i.i.d.}}{\sim} \mathcal{D}(\theta), \; i = 1,\ldots,n \; .\]Often, especially in linear regression models, error terms are independent and identically distributed according to a normal distribution with mean zero and unknown variance:
\[\label{eq:iid-mlr} \varepsilon_i \overset{\mathrm{i.i.d.}}{\sim} \mathcal{N}(0,\sigma^2), \; i = 1,\ldots,n \; .\]- Wikipedia (2024): "Independent and identically distributed random variables"; in: Wikipedia, the free encyclopedia, retrieved on 2024-08-08; URL: https://en.wikipedia.org/wiki/Independent_and_identically_distributed_random_variables#Introduction.
Metadata: ID: D200 | shortcut: iid | author: JoramSoch | date: 2024-08-08, 11:37.