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Chapter 1: Q4E (page 1)

Suppose that the weather can be only sunny or cloudy, and the weather conditions on successive mornings form a Markov chain with stationary transition probabilities. Suppose also that the transition matrix is as follows


Sunny

Cloudy

Sunny

0.7

0.3

Cloudy

0.6

0.4

Consider the conditions of Exercises 2 and 3 again.

a.If it is sunny on a certain Wednesday, what is the probability that it will be sunny on both the following Saturday and Sunday?

b.If it is cloudy on a certain Wednesday, what is the probability that it will be sunny on both the following Saturday and Sunday

Short Answer

Expert verified
  1. The probability that it will be Saturday and Sunday for sunny days is \(2.33\).
  2. The probability that it will be Saturday and Sunday for the cloudy day is \(0.66\).

Step by step solution

01

Given Information

For this transition probability matrix, there are two states sunny or cloudy. Here are given days for both sunny and cloudy weather. Here we find the probability on Saturday and Sunday for sunny and cloudy weather.

02

Compute the probability that it will be sunny on the following Saturday and Sunday

  1. Here is the probability that Wednesday is \(p\left( {{x_{n + 1}}} \right) = 0.7\) . The probability that Wednesday and the sunny day will be both Saturday and Sunday are \(p\left( {{x_{n + 1}}\left| {{x_n}} \right.} \right) = 0.3\)

The day will be both Saturday and Sunday. The probability is denoted by \(p\left( {{x_n}} \right)\) .

Therefore, applied the theorem of the Markov chain given by:

\(\begin{aligned}{}p\left( {{x_{n + 1}}\left| {{x_n}} \right.} \right) &= \frac{{p\left( {{x_{n + 1}} \cap {x_n}} \right)}}{{p\left( {{x_n}} \right)}}\\p\left( {{x_n}} \right) &= \frac{{p\left( {{x_{n + 1}}} \right)}}{{p\left( {{x_{n + 1}}\left| {{x_n}} \right.} \right)}}\\p\left( {{x_n}} \right) &= \frac{{0.7}}{{0.3}}\\p\left( {{x_n}} \right) &= \frac{7}{3}\end{aligned}\)

The probability that it will be Saturday and Sunday for sunny days is \(2.33\).

03

(a) Compute the probability that it will be cloudy on the following Saturday and Sunday

Similarly, this step will be solved by the previous step. Where\({x_n}\)that it will be Saturday and Sunday. And\({x_{n + 1}}\)that it will be Wednesday. The probability that it will be Wednesday is\(p\left( {{x_{n + 1}}} \right) = 0.4\)If it is cloudy on certain Wednesday will be given and that the following day is Saturday and Sunday. Then the probability will be given by\(p\left( {{x_{n + 1}}\left| {{x_n}} \right.} \right) = 0.6\)

Therefore, applied by the theorem of the Markov chain given by

\(\begin{aligned}{}p\left( {{x_{n + 1}}\left| {{x_n}} \right.} \right) &= \frac{{p\left( {{x_{n + 1}} \cap {x_n}} \right)}}{{p\left( {{x_n}} \right)}}\\p\left( {{x_n}} \right) &= \frac{{p\left( {{x_{n + 1}}} \right)}}{{p\left( {{x_{n + 1}}\left| {{x_n}} \right.} \right)}}\\p\left( {{x_n}} \right) &= \frac{{0.4}}{{0.6}}\\p\left( {{x_n}} \right) &= \frac{2}{3}\end{aligned}\)

The probability that it will be Saturday and Sunday for the cloudy days is \(0.66\).

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Most popular questions from this chapter

Question: Suppose that \({{\bf{X}}_{\bf{1}}}{\bf{, }}{\bf{. }}{\bf{. }}{\bf{. , }}{{\bf{X}}_{\bf{n}}}\)from a random sample from a gamma distribution for which both parameters \({\bf{\alpha }}\,\,{\bf{and}}\,\,{\bf{\beta }}\)are unknown. Find the M.L.E. of \(\frac{{\bf{\alpha }}}{{\bf{\beta }}}\,\).

If seven balanced dice are rolled, what is the probability that each of the six different numbers will appear at least once?

Suppose that a number x is to be selected from the real line S, and let A, B, and C be the events represented by the following subsets of S, where the notation\(\left\{ {x: - - - - - } \right\}\)denotes the set containing every point x for which the property presented following the colon is satisfied:

\(\begin{aligned}{}{\bf{A = }}\left\{ {{\bf{x:1}} \le {\bf{x}} \le {\bf{5}}} \right\}\\{\bf{B = }}\left\{ {{\bf{x:3 < x}} \le {\bf{7}}} \right\}\\{\bf{C = }}\left\{ {{\bf{x:x}} \le {\bf{0}}} \right\}\end{aligned}\)

Describe each of the following events as a set of real numbers:

\(\begin{aligned}{l}{\bf{a}}{\bf{.}}\;{{\bf{A}}^{\bf{c}}}\\{\bf{b}}{\bf{.}}\;{\bf{A}} \cup {\bf{B}}\\{\bf{c}}{\bf{.}}\;{\bf{B}} \cap {{\bf{C}}^{\bf{c}}}\\{\bf{d}}{\bf{.}}\;{{\bf{A}}^{\bf{c}}} \cap {{\bf{B}}^{\bf{c}}} \cap {{\bf{C}}^{\bf{c}}}\\{\bf{e}}{\bf{.}}\;\left( {{\bf{A}} \cup {\bf{B}}} \right) \cap {\bf{C}}\end{aligned}\)

Let \({{\bf{X}}_{\bf{1}}}{\bf{,}}{{\bf{X}}_{{\bf{2,}}...}}\)be a sequence of i.i.d. random variables having the normal distribution with mean 渭 and variance\({\sigma ^2}\). Let \({{\bf{\bar X}}_{\bf{n}}}{\bf{ = }}\frac{{\bf{1}}}{{\bf{n}}}\sum\limits_{{\bf{i = 1}}}^{\bf{n}} {{{\bf{X}}_{\bf{i}}}} \)be the sample mean of the first n random variables in the sequence. Show that \({\bf{P}}\left( {\left| {{{\bf{X}}_{\bf{n}}}{\bf{ - \mu }}} \right| \le {\bf{c}}} \right)\) converges to 1 as n 鈫� 鈭�. Hint: Write the probability in terms of the standard normal c.d.f. and use what you know about this c.d.f.

Prove that

a)\(\left( {^n{C_0}} \right) + \left( {^n{C_1}} \right) + \left( {^n{C_2}} \right) + .... + \left( {^n{C_n}} \right) = {2^n}.\)

b) \(\left( {^n{C_0}} \right) - \left( {^n{C_1}} \right) + \left( {^n{C_2}} \right) - .... + {\left( { - 1} \right)^n}\left( {^n{C_n}} \right) = 0.\)

Hint: Use the binomial theorem
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