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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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Question:Suppose thatXandYare random variables such that(X, Y)must belong to the rectangle in thexy-plane containing all points(x, y)for which 0鈮�x鈮�3 and 0鈮�y鈮�4. Suppose also that the joint c.d.f. ofXandYat every point
(x,y) in this rectangle is specified as follows:
\({\bf{F}}\left( {{\bf{x,y}}} \right){\bf{ = }}\frac{{\bf{1}}}{{{\bf{156}}}}{\bf{xy}}\left( {{{\bf{x}}^{\bf{2}}}{\bf{ + y}}} \right)\)
Determine
(a) Pr(1鈮�X鈮�2 and 1鈮�Y鈮�2);
(b) Pr(2鈮�X鈮�4 and 2鈮�Y鈮�4);
(c) the c.d.f. ofY;
(d) the joint p.d.f. ofXandY;
(e) Pr(Y鈮�X).
Suppose that the joint distribution of X and Y is uniform over the region in the\({\bf{xy}}\)plane bounded by the four lines\({\bf{x = - 1,x = 1,y = x + 1}}\)and\({\bf{y = x - 1}}\). Determine (a)\({\bf{Pr}}\left( {{\bf{XY > 0}}} \right)\)and (b) the conditional p.d.f. of Y given that\({\bf{X = x}}\).
Suppose that\({X_1}....{X_n}\)are i.i.d. random variables, each having the following c.d.f.:\(F\left( x \right) = \left\{ \begin{array}{l}0\,\,\,\,\,\,\,\,\,\,\,\,\,\,for\,x \le 0\\1 - {e^{ - x}}\,\,\,for\,x > 0\end{array} \right.\)
Let\({Y_1} = min\left\{ {{X_1},{X_2}..{X_n}} \right\}\)and\({Y_n} = max\left\{ {{X_{1,}}{X_2}..{X_n}} \right\}\)Determine the conditional p.d.f. of\({Y_1}\)given that\({Y_n} = {y_n}\)
Each student in a certain high school was classified according to her year in school (freshman, sophomore, junior, or senior) and according to the number of times that she had visited a certain museum (never, once, or more than once). The proportions of students in the various classifications are given in the following table:
Never once More than once
than once
Freshmen 0.08 0.10 0.04
Sophomores 0.04 0.10 0.04
Juniors 0.04 0.20 0.09
Seniors 0.02 0.15 0.10
a. If a student selected at random from the high school is a junior, what is the probability that she has never visited the museum?
b. If a student selected at random from the high school has visited the museum three times, what is the probability that she is a senior?
Suppose that a point (X, Y) is chosen at random from the disk S defined as follows:
\(S = \left\{ {\left( {x,y} \right) :{{\left( {x - 1} \right)}^2} + {{\left( {y + 2} \right)}^2} \le 9} \right\}.\) Determine (a) the conditional pdf of Y for every given value of X, and (b) \({\rm P}\left( {Y > 0|x = 2} \right)\)
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