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Chapter 3: Q1E (page 100)
Suppose that a random variableXhas the uniform distributionon the integers 10, . . . ,20. Find the probability thatXis even.
The probability that X is even is \(\frac{6}{{11}}\) .
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Question:LetYbe the rate (calls per hour) at which calls arrive at a switchboard. LetXbe the number of calls during at wo-hour period. A popular choice of joint p.f./p.d.f. for(X, Y )in this example would be one like
\({\bf{f}}\left( {{\bf{x,y}}} \right){\bf{ = }}\left\{ \begin{array}{l}\frac{{{{\left( {{\bf{2y}}} \right)}^{\bf{x}}}}}{{{\bf{x!}}}}{{\bf{e}}^{{\bf{ - 3y}}}}\;{\bf{if}}\;{\bf{y > 0}}\;{\bf{and}}\;{\bf{x = 0,1, \ldots }}\\{\bf{0}}\;{\bf{otherwise}}\end{array} \right.\)
a. Verify thatfis a joint p.f./p.d.f. Hint:First, sum overthexvalues using the well-known formula for thepower series expansion of\({{\bf{e}}^{{\bf{2y}}}}\).
b. Find Pr(X=0).
Suppose that a coin is tossed repeatedly in such a way that heads and tails are equally likely to appear on any given toss and that all tosses are independent, with the following exception: Whenever either three heads or three tails have been obtained on three successive tosses, then the outcome of the next toss is always of the opposite type. At time\(n\left( {n \ge 3} \right)\)let the state of this process be specified by the outcomes on tosses\(n - 2\),\(n - 1\)and n. Show that this process is a Markov chain with stationary transition probabilities and construct the transition matrix.
Question:A painting process consists of two stages. In the first stage, the paint is applied, and in the second stage, a protective coat is added. Let X be the time spent on the first stage, and let Y be the time spent on the second stage. The first stage involves an inspection. If the paint fails the inspection, one must wait three minutes and apply the paint again. After a second application, there is no further inspection. The joint pdf.of X and Y is
\(f\left( {x,y} \right) = \left\{ \begin{array}{l}\frac{1}{3}if\,1 < x < 3\,and\,0 < y < 1\\\frac{1}{6}if\,1 < x < 3\,and\,0 < y < 1\,\\0\,\,otherwise.\\\,\end{array} \right.\,\,\)
a. Sketch the region where f (x, y) > 0. Note that it is not exactly a rectangle.
b. Find the marginal p.d.f.鈥檚 of X and Y.
c. Show that X and Y are independent.
Suppose that a random variable X has a uniform distribution on the interval [0, 1]. Determine the p.d.f. of (a)\({{\bf{X}}^{\bf{2}}}\), (b) \({\bf{ - }}{{\bf{X}}^{\bf{3}}}\), and (c) \({{\bf{X}}^{\frac{{\bf{1}}}{{\bf{2}}}}}\).
Suppose that three random variables X1, X2, and X3 have a continuous joint distribution with the following joint p.d.f.:
\({\bf{f}}\left( {{{\bf{x}}_{\bf{1}}}{\bf{,}}{{\bf{x}}_{\bf{2}}}{\bf{,}}{{\bf{x}}_{\bf{3}}}} \right){\bf{ = }}\left\{ {\begin{align}{}{{\bf{c}}\left( {{{\bf{x}}_{\bf{1}}}{\bf{ + 2}}{{\bf{x}}_{\bf{2}}}{\bf{ + 3}}{{\bf{x}}_{\bf{3}}}} \right)}&{{\bf{for0}} \le {{\bf{x}}_{\bf{i}}} \le {\bf{1}}\,\,\left( {{\bf{i = 1,2,3}}} \right)}\\{\bf{0}}&{{\bf{otherwise}}{\bf{.}}}\end{align}} \right.\)
Determine\(\left( {\bf{a}} \right)\)the value of the constant c;
\(\left( {\bf{b}} \right)\)the marginal joint p.d.f. of\({{\bf{X}}_{\bf{1}}}\)and\({{\bf{X}}_{\bf{3}}}\); and
\(\left( {\bf{c}} \right)\)\({\bf{Pr}}\left( {{{\bf{X}}_{\bf{3}}}{\bf{ < }}\frac{{\bf{1}}}{{\bf{2}}}\left| {{{\bf{X}}_{\bf{1}}}{\bf{ = }}\frac{{\bf{1}}}{{\bf{4}}}{\bf{,}}{{\bf{X}}_{\bf{2}}}{\bf{ = }}\frac{{\bf{3}}}{{\bf{4}}}} \right.} \right){\bf{.}}\)
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