Q. 2.5 For any sequence of events E1,聽... [FREE SOLUTION]

91影视

A First Course in Probability

Sheldon M. Ross

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 432 pages

ISBN: 9780321794772

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 $F i F j = 脴$ whenever $i 鈮� j$ ) such that for all $n 鈮� 1$ , ${鈭�}_{1}^{n} F_{i} = {鈭�}_{1}^{n} E_{i}$

Short Answer

Expert verified

We have ${鈭�}_{1}^{n} F_{i} = {鈭�}_{1}^{n} E_{i}$ .

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.

$F_{i} F_{j} = 蠁, i 鈮� j E_{n}; n 鈮� 1 E_{1} 鈯� E_{2} 鈯� . . . 鈯� E_{n} 鈯� E_{n + 1} 鈯� . . . \lim_{n 鈫� 鈭�} E_{n} = {鈭狤}_{i} E_{1} 鈯� E_{2} 鈯� . . . 鈯� E_{n} 鈯� E_{n + 1} \lim_{n 鈫� 鈭�} E_{n} = {鈭〦}_{n}$

$F_{1}, F_{2}, . . . F_{1} F_{2} = 蠁, F_{1} 鈭� F_{2} = 蠁蠁鈯� E_{1} 鈯� E_{2} . . . 鈯� E_{n} 蠁鈯� (E_{1} 鈭� 蠁^{c}) 鈯� (E_{2} 鈭� {E_{1}}^{c}) . . . 鈯� (E_{n} 鈭� E_{n - 1})$

$F_{1} = E_{1} 鈭� 蠁^{c} F_{2} = E_{2} 鈭� {E^{c}}_{1} . . . F_{n} = E_{n} 鈭� E_{n - 1}$

$F_{1} F_{2} = (E_{1} 鈭� 蠁^{c}) 鈭� (E_{2} 鈭� {E_{1}}^{c}) = 蠁鈭� 蠁^{c} 鈭� E_{2} = E_{1} 鈭� E_{1} = 蠁$

$F_{i} F_{j} = (E_{i} 鈭� {E^{c}}_{i - 1}) 鈭� (E_{j} 鈭� {E^{c}}_{j - 1}) = E_{i} 鈭� {E^{c}}_{j - 1} = 蠁$

{鈭狥}_{i} = (E_{1} 鈭� 蠁^{c}) 鈭� (E 2 鈭� {E_{1}}^{c}) 鈭� . . . 鈭� (E_{n} 鈭� {E_{n - 1}}^{c}) = {鈭狤}_{i}

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91影视

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 $F i F j = 脴$ whenever $i 鈮� j$ ) such that for all $n 鈮� 1$ , ${鈭�}_{1}^{n} F_{i} = {鈭�}_{1}^{n} E_{i}$

Short Answer

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Explanation.

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91影视

For any sequence of events E1,E2,...,define a new sequenceF1,F2,...of disjoint events (that is, events such that FiFj=脴whenever i鈮�j) such that for alln鈮�1, 鈭�1nFi=鈭�1nEi

Short Answer

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Given Information.

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 $F i F j = 脴$ whenever $i 鈮� j$ ) such that for all $n 鈮� 1$ , ${鈭�}_{1}^{n} F_{i} = {鈭�}_{1}^{n} E_{i}$