Chapter 2: Q. 2.5 (page 52)

For any sequence of events $E 1, E 2, . . .,$ define a new sequence $F 1, F 2, . . .$ of disjoint events (that is, events such that $F i F j = Ø$ whenever $i \neq j$ ) such that for all $n \geq 1$ , $\cup_{1}^{n} F_{i} = \cup_{1}^{n} E_{i}$

Short Answer

Expert verified

We have $\cup_{1}^{n} F_{i} = \cup_{1}^{n} E_{i}$ .

Step by step solution

Given Information.

For any sequence of events $E 1, E 2, . . .,$ define a new sequence localid="1649078122634" $F 1, F 2, . . .$ of disjoint events.

Explanation.

$F_{i} F_{j} = φ, i \neq j E_{n}; n \geq 1 E_{1} \subset E_{2} \subset . . . \subset E_{n} \subset E_{n + 1} \subset . . . \lim_{n \to \infty} E_{n} = \cupE_{i} E_{1} \supset E_{2} \supset . . . \supset E_{n} \supset E_{n + 1} \lim_{n \to \infty} E_{n} = \capE_{n}$

$F_{1}, F_{2}, . . . F_{1} F_{2} = φ, F_{1} \cap F_{2} = φ φ \subset E_{1} \subset E_{2} . . . \subset E_{n} φ \subset (E_{1} \cap φ^{c}) \subset (E_{2} \cap {E_{1}}^{c}) . . . \subset (E_{n} \cap E_{n - 1})$

$F_{1} = E_{1} \cap φ^{c} F_{2} = E_{2} \cap {E^{c}}_{1} . . . F_{n} = E_{n} \cap E_{n - 1}$

$F_{1} F_{2} = (E_{1} \cap φ^{c}) \cap (E_{2} \cap {E_{1}}^{c}) = φ \cap φ^{c} \cap E_{2} = E_{1} \cap E_{1} = φ$

$F_{i} F_{j} = (E_{i} \cap {E^{c}}_{i - 1}) \cap (E_{j} \cap {E^{c}}_{j - 1}) = E_{i} \cap {E^{c}}_{j - 1} = φ$

\cupF_{i} = (E_{1} \cap φ^{c}) \cup (E 2 \cap {E_{1}}^{c}) \cup . . . \cup (E_{n} \cap {E_{n - 1}}^{c}) = \cupE_{i}

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For any sequence of events $E 1, E 2, . . .,$ define a new sequence $F 1, F 2, . . .$ of disjoint events (that is, events such that $F i F j = Ø$ whenever $i \neq j$ ) such that for all $n \geq 1$ , $\cup_{1}^{n} F_{i} = \cup_{1}^{n} E_{i}$

Short Answer

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Given Information.

Explanation.

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For any sequence of events E1,E2,...,define a new sequenceF1,F2,...of disjoint events (that is, events such that FiFj=Øwhenever i≠j) such that for alln≥1, ∪1nFi=∪1nEi

Short Answer

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Explanation.

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For any sequence of events $E 1, E 2, . . .,$ define a new sequence $F 1, F 2, . . .$ of disjoint events (that is, events such that $F i F j = Ø$ whenever $i \neq j$ ) such that for all $n \geq 1$ , $\cup_{1}^{n} F_{i} = \cup_{1}^{n} E_{i}$