Proof: Self-independence of random event

Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Probability ▷ Self-independence

Theorem: Let $E$ be a random event. Then, $E$ is independent of itself, if and only if its probability is zero or one:

\[\label{eq:ind-self} E \text{ self-independent} \quad \Leftrightarrow \quad P(E) = 0 \quad \text{or} \quad P(E) = 1 \; .\]

Proof: According to the definition of statistical independence, it must hold that:

\[\label{eq:ind} \begin{split} P(E,E) &= P(E) \cdot P(E) \\ P(E) &= \left( P(E) \right)^2 \; . \end{split}\]

For $0 \leq P(E) \leq 1$, this is only fulfilled, if

\[\label{eq:ind-self-qed} P(E) = 0 \quad \text{or} \quad P(E) = 1 \; .\]

Both is possible, since the lower bound of probability is zero and the upper bound of probability is one.

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Sources:

• Wikipedia (2024): "Independence (probability theory)"; in: Wikipedia, the free encyclopedia, retrieved on 2024-09-20; URL: https://en.wikipedia.org/wiki/Independence_(probability_theory)#Self-independence.
• Soch, Joram (2023): "Suppose A is an event. Can A be independent of itself?"; in: X, Aug 7, 2023, 03:59 PM; URL: https://x.com/JoramSoch/status/1688550557034651648.

Metadata: ID: P470 | shortcut: ind-self | author: JoramSoch | date: 2024-09-20, 14:04.