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Chapter 4: Q4.4-3E (page 240)
Suppose thatXis a random variable for whichE(X)=1, \(E\left( {{X^2}} \right) = 2\), and \(E\left( {{X^3}} \right) = 5\). Find the value of the thirdcentral moment ofX.
The third central moment of X is 1.
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Prove the following extension of Theorem 4.4.1: If \(E\left( {{{\left| X \right|}^a}} \right) < \infty \) for some positive number a, then \(E\left( {{{\left| X \right|}^b}} \right) < \infty \) for every positive number \(b < a\). Give the proof for the case in which X has a discrete distribution.
Suppose that a person's score X on a mathematics aptitude test is a number in the interval\(\left( {0,1} \right)\)and that his score Y on a music aptitude test is also a number in the interval\(\left( {0,1} \right)\)Suppose also that in the population of all college students in the United States, the scores X and Y are distributed in accordance with the following joint p.d.f:
\(f\left( {x,y} \right) = \left\{ \begin{align}\frac{2}{5}\left( {2x + 3y} \right)\;\;\;\;\;\;\;for\,0 \le x \le 1\,and0 \le x \le 1\\0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;otherwise\end{align} \right.\)
a. If a college student is selected randomly, what predicted value of his score on the music test has the smallest M.S.E.?
b. What predicted value of his score on the mathematics test has the smallest M.A.E.?
Consider again the conditions of Exercise 2, but suppose now that X has a discrete distribution with c.d.f.\(F\left( x \right)\)F (x),rather than a continuous distribution. Show that the conclusion of Exercise 2 still holds
Suppose that on each play of a certain game a gambleris equally likely to win or to lose. Suppose that when hewins, his fortune is doubled and that when he loses, his fortune is cut in half. If he begins playing with a given
fortunec, what is the expected value of his fortune afternindependent plays of the game?
Suppose that three random variables\({{\bf{X}}_{\bf{1}}}{\bf{,}}{{\bf{X}}_{\bf{2}}}{\bf{,}}{{\bf{X}}_{\bf{3}}}\)froma random sample from a distribution for which the meanis 5. Determine the value of
\({\bf{E}}\left( {{\bf{2}}{{\bf{X}}_{\bf{1}}}{\bf{ - 3}}{{\bf{X}}_{\bf{2}}}{\bf{ + }}{{\bf{X}}_{\bf{3}}}{\bf{ - 4}}} \right)\).
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