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Question 2: A laboratory animal may eat any one of three foods each day. Laboratory records show that if the animal chooses one food on one trial, it will chose the same food on the next trial with a probability of 50%, and it will choose the other foods on the next trial with equal probabilities of 25%.

a. What is the stochastic matrix for this situation?

b. If the animal chooses food #1 on an initial trial, what is the probability that it will choose food #2 on the second trial after the initial trial?

Short Answer

Expert verified
  1. The stochastic matrix is \(P = \left( {\begin{array}{*{20}{c}}{.5}&{.25}&{.25}\\{.25}&{.5}&{.25}\\{.25}&{.25}&{.5}\end{array}} \right)\).
  2. The probability that the animal will choose food #2 after the initial trial is 0.3125.

Step by step solution

01

Determine the stochastic matrix

a)

Let the food be labeled 1, 2, and 3.

The following table represents the animal's behavior.

From:

To:

1

2

3

.5

.25

.25

.25

.5

.25

.25

.25

.5

1

2

3

From this, the stochastic matrix is \(P = \left( {\begin{array}{*{20}{c}}{.5}&{.25}&{.25}\\{.25}&{.5}&{.25}\\{.25}&{.25}&{.5}\end{array}} \right)\).

02

Determine the probability that will choose food #2 after the initial trial

b)

After the initial trial, there are two trials. The initial state vector is \({{\mathop{\rm x}\nolimits} _0} = \left( {\begin{array}{*{20}{c}}1\\0\\0\end{array}} \right)\).

Compute \({{\mathop{\rm x}\nolimits} _2}\) as shown below:

\(\begin{array}{c}{{\mathop{\rm x}\nolimits} _1} = P{{\mathop{\rm x}\nolimits} _0}\\ = \left( {\begin{array}{*{20}{c}}{.5}&{.25}&{.25}\\{.25}&{.5}&{.25}\\{.25}&{.25}&{.5}\end{array}} \right)\left( {\begin{array}{*{20}{c}}1\\0\\0\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{0.5 + 0 + 0}\\{0.25 + 0 + 0}\\{0.25 + 0 + 0}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{0.5}\\{0.25}\\{0.25}\end{array}} \right)\end{array}\)

\(\begin{array}{c}{{\mathop{\rm x}\nolimits} _2} = P{{\mathop{\rm x}\nolimits} _1}\\ = \left( {\begin{array}{*{20}{c}}{.5}&{.25}&{.25}\\{.25}&{.5}&{.25}\\{.25}&{.25}&{.5}\end{array}} \right)\left( {\begin{array}{*{20}{c}}{.5}\\{.25}\\{.25}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{0.25 + 0.0625 + 0.0625}\\{0.25 + 0.125 + 0.0625}\\{0.125 + 0.0625 + 0.125}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{0.375}\\{0.3125}\\{0.3125}\end{array}} \right)\end{array}\)

Thus, the probability that the animal will choose food #2 after the initial trial is 0.3125.

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Most popular questions from this chapter

Exercises 23-26 concern a vector space V, a basis \(B = \left\{ {{{\bf{b}}_{\bf{1}}},....,{{\bf{b}}_n}\,} \right\}\)and the coordinate mapping \({\bf{x}} \mapsto {\left( {\bf{x}} \right)_B}\).

Given vectors, \({u_{\bf{1}}}\),….,\({u_p}\) and w in V, show that w is a linear combination of \({u_{\bf{1}}}\),….,\({u_p}\) if and only if \({\left( w \right)_B}\) is a linear combination of vectors \({\left( {{{\bf{u}}_{\bf{1}}}} \right)_B}\),….,\({\left( {{{\bf{u}}_p}} \right)_B}\).

Which of the subspaces \({\rm{Row }}A\) , \({\rm{Col }}A\), \({\rm{Nul }}A\), \({\rm{Row}}\,{A^T}\) , \({\rm{Col}}\,{A^T}\) , and \({\rm{Nul}}\,{A^T}\) are in \({\mathbb{R}^m}\) and which are in \({\mathbb{R}^n}\) ? How many distinct subspaces are in this list?.

Question: Exercises 12-17 develop properties of rank that are sometimes needed in applications. Assume the matrix \(A\) is \(m \times n\).

  1. Show that if \(B\) is \(n \times p\), then rank\(AB \le {\mathop{\rm rank}\nolimits} A\). (Hint: Explain why every vector in the column space of \(AB\) is in the column space of \(A\).
  2. Show that if \(B\) is \(n \times p\), then rank\(AB \le {\mathop{\rm rank}\nolimits} B\). (Hint: Use part (a) to study rank\({\left( {AB} \right)^T}\).)

In Exercise 17, Ais an \(m \times n\] matrix. Mark each statement True or False. Justify each answer.

17. a. The row space of A is the same as the column space of \({A^T}\].

b. If B is any echelon form of A, and if B has three nonzero rows, then the first three rows of A form a basis for Row A.

c. The dimensions of the row space and the column space of A are the same, even if Ais not square.

d. The sum of the dimensions of the row space and the null space of A equals the number of rows in A.

e. On a computer, row operations can change the apparent rank of a matrix.

Is it possible for a nonhomogeneous system of seven equations in six unknowns to have a unique solution for some right-hand side of constants? Is it possible for such a system to have a unique solution for every right-hand side? Explain.

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