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Distribution of a finite random variable

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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 marked, marked. Their numerical codes, in the same order, are 0, 1

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

At the third stage, the possible outcomes are not marked, marked. 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 marked, not marked, not marked)	1/8
(not marked, not marked, marked)	1/8
(not marked, marked, not marked)	1/8
(not marked, marked, marked)	1/8
(marked, not marked, not marked)	1/8
(marked, not marked, marked)	1/8
(marked, marked, not marked)	1/8
(marked, marked, marked)	1/8

Define \(X\) as the number of stages whose outcome is marked

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 \(X\)?

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 marked, not marked, not marked)	0
(not marked, not marked, marked)	0
(not marked, marked, not marked)	0
(not marked, marked, marked)	0
(marked, not marked, not marked)	1
(marked, not marked, marked)	1
(marked, marked, not marked)	1
(marked, marked, marked)	1

Apply the stated rule to every ordered outcome

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

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 marked, not marked, not marked)	0
(not marked, not marked, marked)	1
(not marked, marked, not marked)	0
(not marked, marked, marked)	1
(marked, not marked, not marked)	0
(marked, not marked, marked)	1
(marked, marked, not marked)	0
(marked, marked, marked)	1

(b)2 marks

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

Frequency for X = 0 Frequency for X = 3

What is \(Pr(X\in \{1,\,2,\,3\})\)? Give an exact fraction or a terminating decimal when possible

\(Pr(X\in \{1,\,2,\,3\})\)

Format: exact fraction