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Chapter 4: Q4.1-6E (page 216)
Suppose that a random variable X has a continuous distribution with the p.d.f. has given in Example 4.1.6. Find the expectation of 1/X
\(E\left( {\frac{1}{X}} \right) = 2\)
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Suppose that X, Y , and Z are nonnegative random variables such that\({\rm P}\left( {X + Y + Z < 1.3} \right) = 1\). Show that X, Y , and Z cannot possibly have a joint distribution under which each of their marginal distributions is the uniform distribution on the interval\(\left( {0,1} \right)\).
Suppose that the distribution of X is symmetric around a point m. Prove that m is a median of X.
Suppose that a random variable X has a discrete distribution for which the p.f. is as follows:
\(f\left( x \right) = \left\{ {\begin{aligned}{{}{}}{cx}&{{\rm{for }}x = 1,2,3,4,5,6}\\0&{{\rm{otherwise}}}\end{aligned}} \right.\)
Determine all the medians of this distribution.
Consider again the situation described in Example 4.7.8. Compute the M.S.E. when using \({\bf{E}}\left( {{\bf{P|x}}} \right)\)to predictPafter observingX=18. How much smaller is this than the marginal M.S.E. 1/12?
Suppose that a particle starts at the origin of the whole line and moves along the line in jumps of one unit. For each jump, the probability is p (0 鈮� p 鈮� 1) that the particle will jump one unit to the left, and the probability is 1 鈭� p that the particle will jump one unit to the right. Find the expected value of the position of the particle after n jumps
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