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Chapter 3: Q16E (page 117)
Find the quantile function for the distribution in Example 3.3.1.
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Suppose that a person鈥檚 score X on a mathematics aptitude test is a number between 0 and 1, and that his score Y on a music aptitude test is also a number between 0 and 1. Suppose further that in the population of all college students in the United States, the scores X and Y are distributed according to the following joint pdf:
\(f\left( {x,y} \right)\left\{ \begin{aligned}\frac{2}{5}\left( {2x + 3y} \right)for0 \le x \le 1 and 0 \le y \le 1\\0 otherwise\end{aligned} \right.\)
a. What proportion of college students obtain a score greater than 0.8 on the mathematics test?
b. If a student鈥檚 score on the music test is 0.3, what is the probability that his score on the mathematics test will be greater than 0.8?
c. If a student鈥檚 score on the mathematics test is 0.3, what is the probability that his score on the music test will be greater than 0.8?
Consider the Markov chain in Example 3.10.2 with initial
probability vector \(v = \left( {\frac{1}{2},\frac{1}{2}} \right)\) Where \(p = \left[ {\begin{array}{*{20}{c}}{\frac{1}{3}}&{\frac{2}{3}}\\{\frac{1}{3}}&{\frac{1}{3}}\end{array}} \right]\)
a.Find the probability vector specifying the probabilities
of the states at timen=2.
b.Find the two-step transition matrix
Question:Suppose that two persons make an appointment to meet between 5 p.m. and 6 p.m. at a certain location, and they agree that neither person will wait more than 10 minutes for the other person. If they arrive independently at random times between 5 p.m. and 6 p.m. what is the probability that they willmeet?
Suppose that a random variableXhas the uniform distributionon the integers 10, . . . ,20. Find the probability thatXis even.
For the conditions of Exercise 9, determine the value of \({\bf{Pr}}\left( {{{\bf{Y}}_{\bf{1}}} \le {\bf{0}}{\bf{.1}}\;{\bf{and}}\;{\bf{Y}}_{\bf{n}}^{} \ge {\bf{0}}{\bf{.8}}} \right)\).
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