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Chapter 4: Q4.4-2E (page 240)
IfXhas the uniform distribution on the interval [a, b], write a formula for every even central moment ofX.
The fifth central moment of X is 0.
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Suppose that the pair(X, Y )is uniformly distributed on the interior of a circle of radius 1. Compute蚁(齿,驰).
Suppose that X and Y have a continuous joint distribution for which the joint p.d.f. is as follows:
\({\bf{f}}\left( {{\bf{x,y}}} \right){\bf{ = }}\left\{ {\begin{align}{}{{\bf{x + y}}}&{{\bf{for}}\,\,{\bf{0}} \le {\bf{x}} \le {\bf{1}}\,\,{\bf{and}}\,\,{\bf{0}} \le {\bf{y}} \le {\bf{1,}}}\\{\bf{0}}&{{\bf{otherwise}}{\bf{.}}}\end{align}} \right.\).
Find\({\bf{E}}\left( {{\bf{Y}}\left| {\bf{X}} \right.} \right)\)and\({\bf{Var}}\left( {{\bf{Y}}\left| {\bf{X}} \right.} \right)\).
Let X have the discrete uniform distribution on the integers \(1, \ldots \ldots ,n\).Compute the variance of X.
Hint: Youmay wish to use the formula \(\sum\limits_{k = 1}^n {{k^2}} = \frac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{6}\)
Suppose that a point is chosen at random on a stick of unit length and that the stick is broken into two pieces at that point. Find the expected value of the size of the longer piece.
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?
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