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Chapter 3: Q9E (page 99)

An individual named Claudius is located at the point 0 in the accompanying diagram. Using an appropriate randomization device (such as a
tetrahedral die, one having four sides), Claudius first moves to one of the four locations B1, B2, B3, B4. Once at one of these locations, another randomization device is used to decide whether Claudius next returns to 0 or next visits one of the other two adjacent points. This process then continues; after each move, another move to one of the (new) adjacent points is determined by tossing an appropriate die or coin.

a. Let X = the number of moves that Claudius makes before first returning to 0. What are possible values of X? Is X discrete or continuous?

b. If moves are allowed also along the diagonal paths connecting 0 to A1, A2, A3, and A4, respectively, answer the questions in part (a).

Short Answer

Expert verified

a. The possible values of X are\(X = \left\{ {2,4,6, \ldots } \right\}\). Discrete

b. The possible values of X are\(X = \left\{ {2,3,4,5, \ldots } \right\}\). Discrete

Step by step solution

01

Given information

There is a person called Claudius located at the point zero and the experiment is performed in which he has to move to any of the four location B1, B2, B3 and B4, and the next move is either return to zero or move to other two adjacent point is observed.

02

Determine the possible values of random variable X

a.

Let us suppose Claudius make the second move and return to zero then this the minimum number of moves that Claudius make before returning to zero is two.

If in the second move he moves to any of the other two adjacent points then he has to make two more moves before returning to zero is 4.

In continuation to the above possible values of Xis given as,

\(X = \left\{ {2,4,6, \ldots } \right\}\)

As the possible values of X is countable and finite then this is the discrete random variable.

03

Determine the possible values of X when moves are also allowed along the diagonal paths

b.

In this case if moves are allowed diagonally then the third move is either return to point zero or move to other adjacent points.

Claudius from its location at point zero initially make his first move to any of the four locations B1, B2, B3 and B4 then make the second move and return to zero then this the minimum number of movies that Claudius make before returning to zero is two.

If in the second move he moves to any of the other two adjacent points, then he has to make another move either returning to point 0 or move to other adjacent points.

If he makes a move to return to 0 then the number of moves will be 3.

If he makes a move to return in his fourth move then the total number of moves will be 5.

In continuation to the above possible values of X is given as,

\(X = \left\{ {2,3,4,5, \ldots } \right\}\)

As the possible values of X is countable and finite then this is the discrete random variable.

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