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Distribution of a finite random variable

Question 1 of 8

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The sample space \(S\) contains the ordered outcomes of a finite 3-stage experiment

At the first stage, the possible outcomes are not included, included. Their numerical codes, in the same order, are 0, 1

At the second stage, the possible outcomes are not included, included. Their numerical codes, in the same order, are 0, 1

At the third stage, the possible outcomes are not included, included. Their numerical codes, in the same order, are 0, 1

The exact probabilities of the ordered outcomes are given below. Outcomes from successive stages remain ordered

The listed probabilities are for the ordered outcomes
Ordered outcome	Exact probability
(not included, not included, not included)	1/8
(not included, not included, included)	1/8
(not included, included, not included)	1/8
(not included, included, included)	1/8
(included, not included, not included)	1/8
(included, not included, included)	1/8
(included, included, not included)	1/8
(included, included, included)	1/8

Define \(X\) as the number of stages whose outcome is not included

The possible values of \(X\) are \(\{0,\,1,\,2,\,3\}\)

(a)1 mark

Which option gives the outcome-to-value mapping required by the definition of \(X\)?

Apply the stated rule to every ordered outcome

The proposed values are listed for the ordered outcomes
Ordered outcome	Value of X
(not included, not included, not included)	3
(not included, not included, included)	2
(not included, included, not included)	2
(not included, included, included)	1
(included, not included, not included)	2
(included, not included, included)	1
(included, included, not included)	1
(included, included, included)	0

Use only the last-stage numerical code as the value of X

The proposed values are listed for the ordered outcomes
Ordered outcome	Value of X
(not included, not included, not included)	0
(not included, not included, included)	1
(not included, included, not included)	0
(not included, included, included)	1
(included, not included, not included)	0
(included, not included, included)	1
(included, included, not included)	0
(included, included, included)	1

Use only the first-stage numerical code as the value of X

The proposed values are listed for the ordered outcomes
Ordered outcome	Value of X
(not included, not included, not included)	0
(not included, not included, included)	0
(not included, included, not included)	0
(not included, included, included)	0
(included, not included, not included)	1
(included, not included, included)	1
(included, included, not included)	1
(included, included, included)	1

(b)2 marks

Complete the missing entries in the distribution of \(X\)

Frequency for X = 0 Frequency for X = 3

Calculate the exact probability for the listed values of X: \(Pr(X\in \{0,\,3\})\)

\(Pr(X\in \{0,\,3\})\)

Format: exact fraction