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Chapter 3: Q8E (page 99)

Each time a component is tested, the trial is a success (S) or failure (F). Suppose the component is tested repeatedly until a success occurs on three consecutive trials. Let Y denote the number of trials necessary to achieve this. List all outcomes corresponding to the five smallest possible values of Y, and state which Y value is associated with each one.

Short Answer

Expert verified

Y

Outcome

3

\(\left\{ {SSS} \right\}\)

4

\(\left\{ {FSSS} \right\}\)

5

\(\left\{ {FFSSS,SFSSS} \right\}\)

6

\(\left\{ {FFFSSS,SSFSSS,FSFSSS,SFFSSS} \right\}\)

7

\(\left\{ \begin{array}{l}FFFFSSS,FSSFSSS,FFSFSSS,FSFFSSS,\\SFFFSSS,SSFFSSS,SFSFSSS\end{array} \right\}\)

Step by step solution

01

Given information

The random variable Y is defined as the variable that represents the number of trials necessary to achieve the trial鈥檚 success.

The experiment terminates as soon as a success occurs on three consecutive trials.

02

Determine the possibleY values

Let the trial鈥檚 success is denoted by S and the trial鈥檚 failure is denoted by F.

The random variableY can only take on positive values:\(3,4,5,6,7 \ldots \)as Y counts the three-consecutive success in number of trials.

Therefore, the rv is defined by:

\(\left( {Y:Y \in \left\{ {3,4,5,6, \ldots } \right\}} \right)\)

The random variable Y takes infinite set of values in this experiment.

03

List five possible outcomes with their associated values of Y

Here, Y is defined as the variable of the number of trials required to achieve success and it terminated when success occurs on three consecutive trials.

The outcome of three consecutive successes occurs in at least three trials is\(\left\{ {SSS} \right\}\).

Similarly, the outcome of three consecutive success occur in 4 trials is\(\left\{ {FSSS} \right\}\).

Similarly, the outcome of three consecutive success occur in 5 trials are\(\left\{ {FFSSS,SFSSS} \right\}\).

The following table represent the possible outcomes and their associated values of Y.

Y

Outcome

3

\(\left\{ {SSS} \right\}\)

4

\(\left\{ {FSSS} \right\}\)

5

\(\left\{ {FFSSS,SFSSS} \right\}\)

6

\(\left\{ {FFFSSS,SSFSSS,FSFSSS,SFFSSS} \right\}\)

7

\(\left\{ \begin{array}{l}FFFFSSS,FSSFSSS,FFSFSSS,FSFFSSS,\\SFFFSSS,SSFFSSS,SFSFSSS\end{array} \right\}\)

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Most popular questions from this chapter

Let X have a Poisson distribution with parameter \({\rm{\mu }}\). Show that E(X) =\({\rm{\mu }}\) directly from the definition of expected value. (Hint: The first term in the sum equals \({\rm{0}}\), and then x can be cancelled. Now factor out \({\rm{\mu }}\) and show that what is left sums to \({\rm{1}}\).)

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