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Chapter 4: Q3E (page 191)

Question 3: On any given day, a student is either healthy or ill. Of the students who are healthy today, 95% will be healthy tomorrow. Of the students who are ill today, 55% will still be ill tomorrow.

a. What is the stochastic matrix for this situation?

b. Suppose 20% of the students are ill on Monday. What fraction or percentage of the students are likely to be ill on Tuesday? On Wednesday?

c. If a student is well today, what is the probability that he or she will be well two days from now?

Short Answer

Expert verified
  1. The stochastic matrix is \(P = \left( {\begin{array}{*{20}{c}}{.95}&{.45}\\{.05}&{.55}\end{array}} \right)\).
  2. 15% of the students are ill on Tuesday, and 12.5% of the students are ill on Wednesday.
  1. The probability that the students will be well two days from now is 0.925.

Step by step solution

01

Determine the stochastic matrix

a)

Let \(H\) represent a healthy student and \(I\) represent an ill student.

The following table represents the student鈥檚 conditions.

From:

To:

H

I

.95

.45

H

.05

.55

I

Therefore, the stochastic matrix is \(P = \left( {\begin{array}{*{20}{c}}{.95}&{.45}\\{.05}&{.55}\end{array}} \right)\).

02

Determine the percentage of students who are likely to be ill on Tuesday

b)

Theorem 18states that if \(P\) is a \(n \times n\) regular stochastic matrix,then \(P\) has a unique steady-state vector \({\mathop{\rm q}\nolimits} \).Further, if \({{\mathop{\rm x}\nolimits} _0}\) is any initial state and \({{\mathop{\rm x}\nolimits} _{k + 1}} = P{{\mathop{\rm x}\nolimits} _k}\) for \(k = 0,1,2,....\) then the Markov chain \(\left\{ {{{\mathop{\rm x}\nolimits} _k}} \right\}\) converges to \({\mathop{\rm q}\nolimits} \)as \(k \to \infty \).

The initial state vector is \({{\mathop{\rm x}\nolimits} _0} = \left( {\begin{array}{*{20}{c}}{.8}\\{.2}\end{array}} \right)\) because 20% of the students are ill on Monday.

Compute \({{\mathop{\rm x}\nolimits} _1}\) to find Tuesday鈥檚 percentage.

\(\begin{array}{c}{{\mathop{\rm x}\nolimits} _1} = P{{\mathop{\rm x}\nolimits} _0}\\ = \left( {\begin{array}{*{20}{c}}{.95}&{.45}\\{.05}&{.55}\end{array}} \right)\left( {\begin{array}{*{20}{c}}{.8}\\{.2}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{0.76 + 0.09}\\{0.04 + 0.11}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{0.85}\\{0.15}\end{array}} \right)\end{array}\)

Thus, 15% of the students are ill on Tuesday.

03

Determine the percentage of students likely to be ill on Wednesday

Compute \({{\mathop{\rm x}\nolimits} _1}\) to find Tuesday鈥檚 percentage.

\(\begin{array}{c}{{\mathop{\rm x}\nolimits} _2} = P{{\mathop{\rm x}\nolimits} _1}\\ = \left( {\begin{array}{*{20}{c}}{.95}&{.45}\\{.05}&{.55}\end{array}} \right)\left( {\begin{array}{*{20}{c}}{0.85}\\{0.15}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{0.8075 + 0.0675}\\{0.0425 + 0.0825}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{0.875}\\{0.125}\end{array}} \right)\end{array}\)

Hence, 12.5% of the students are ill on Wednesday.

04

Determine the probability that the student will be well in two days

c)

Suppose the student is well today, then the initial state vector is \({{\mathop{\rm x}\nolimits} _0} = \left( {\begin{array}{*{20}{c}}1\\0\end{array}} \right)\).

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

\(\begin{array}{c}{{\mathop{\rm x}\nolimits} _1} = P{{\mathop{\rm x}\nolimits} _0}\\ = \left( {\begin{array}{*{20}{c}}{.95}&{.45}\\{.05}&{.55}\end{array}} \right)\left( {\begin{array}{*{20}{c}}1\\0\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{0.95 + 0}\\{0.05 + 0}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{0.95}\\{0.05}\end{array}} \right)\end{array}\)

\(\begin{array}{c}{{\mathop{\rm x}\nolimits} _2} = P{{\mathop{\rm x}\nolimits} _1}\\ = \left( {\begin{array}{*{20}{c}}{.95}&{.45}\\{.05}&{.55}\end{array}} \right)\left( {\begin{array}{*{20}{c}}{0.95}\\{0.05}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{0.9025 + 0.0225}\\{0.0475 + 0.0275}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{0.925}\\{0.075}\end{array}} \right)\end{array}\)

Thus, the probability that the students will be well two days from now is 0.925.

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Most popular questions from this chapter

In Exercises 21 and 22, mark each statement True or False. Justify

each answer.

22. a.A linearly independent set in a subspace H is a basis for H.

b. If a finite set S of nonzero vectors spans a vector space V, then some subset of S is a basis for V.

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e. If B is an echelon form of a matrix A, then the pivot columns of B form a basis for Col A.

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

Show that the coordinate mapping is onto \({\mathbb{R}^n}\). That is, given any y in \({\mathbb{R}^n}\), with entries \({y_{\bf{1}}}\),鈥�.,\({y_n}\), produce u in V such that \({\left( {\bf{u}} \right)_B} = y\).

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  2. The correspondence\({\left( {\mathop{\rm x}\nolimits} \right)_B} \mapsto {\mathop{\rm x}\nolimits} \)is called coordinate mapping.
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