Chapter 3: Q18E (page 202)
Return to Example 3.10.13. Prove that the stationary distributions described there are the only stationary distributions for that Markov chain.
Proved
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In a large collection of coins, the probability X that a head will be obtained when a coin is tossed varies from one coin to another, and the distribution of X in the collection is specified by the following p.d.f.:
\({{\bf{f}}_{\bf{1}}}\left( {\bf{x}} \right){\bf{ = }}\left\{ {\begin{align}{}{{\bf{6x}}\left( {{\bf{1 - x}}} \right)}&{{\bf{for}}\,{\bf{0 < x < 1}}}\\{\bf{0}}&{{\bf{otherwise}}}\end{align}} \right.\)
Suppose that a coin is selected at random from the collection and tossed once, and that a head is obtained. Determine the conditional p.d.f. of X for this coin.
Suppose that three boys A, B, and C are throwing a ball from one to another. Whenever A has the ball, he throws it to B with a probability of 0.2 and to C with a probability of 0.8. Whenever B has the ball, he throws it to A with a probability of 0.6 and to C with a probability of 0.4. Whenever C has the ball, he is equally likely to throw it to either A or B.
a. Consider this process to be a Markov chain and construct the transition matrix.
b. If each of the three boys is equally likely to have the ball at a certain time n, which boy is most likely to have the ball at time\(n + 2\).
Let Xbe a random variable with the p.d.f. specified in Example 3.2.6. Compute Pr(X≤8/27).
Suppose that the p.d.f. of a random variable X is as
follows:\(f\left( x \right) = \left\{ \begin{array}{l}\frac{1}{2}x\,\,\,\,\,\,\,\,for\,0 < x < 2\\0\,\,\,\,\,\,\,\,\,\,\,\,otherwise\end{array} \right.\)
Also, suppose that \(Y = X\left( {2 - X} \right)\) Determine the cdf and the pdf of Y .
Suppose that \({{\bf{X}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{X}}_{\bf{n}}}\) form a random sample of sizen from the uniform distribution on the interval [0, 1] andthat \({{\bf{Y}}_{\bf{n}}}{\bf{ = max}}\left( {{{\bf{X}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{X}}_{\bf{n}}}} \right)\). Find the smallest value of \({\bf{n}}\)such that\({\bf{Pr}}\left( {{{\bf{Y}}_{\bf{n}}} \ge {\bf{0}}{\bf{.99}}} \right) \ge {\bf{0}}{\bf{.95}}\).
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