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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 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
Suppose that three random variables \({{\bf{X}}_{\bf{1}}}{\bf{,}}{{\bf{X}}_{\bf{2}}}{\bf{,}}{{\bf{X}}_{\bf{3}}}\)from a random sample from the uniform distribution on theinterval [0,1]. Determine the value of\({\bf{E}}\left[ {{{\left( {{{\bf{X}}_{\bf{1}}}{\bf{ - 2}}{{\bf{X}}_{\bf{2}}}{\bf{ + }}{{\bf{X}}_{\bf{3}}}} \right)}^{\bf{2}}}} \right]\).
Let X have the binomial distribution with parameters n and p. Let Y have the binomial distribution with parameters n and \(1 - p\). Prove that the skewness of Y is the negative of the skewness of X. Hint: Let \(Z = n - X\) and show that Z has the same distribution as Y.
Suppose that a fair coin is tossed repeatedly until a head is obtained for the first time.
(a) What is the expected number of tosses that will be required?
(b) What is the expected number of tails that will be obtained before the first head is obtained?
Suppose that a point\({{\bf{X}}_{\bf{1}}}\)is chosen from the uniform distribution on the interval\(\left( {{\bf{0,1}}} \right)\)and that after the value\({{\bf{X}}_{\bf{1}}}{\bf{ = }}{{\bf{x}}_{\bf{1}}}\)is observed, a point\({{\bf{X}}_{\bf{2}}}\)is chosen from a uniform distribution on the interval\(\left( {{{\bf{x}}_{\bf{1}}}{\bf{,1}}} \right)\). Suppose further that additional variables\({{\bf{X}}_{\bf{3}}}{\bf{,}}{{\bf{X}}_{\bf{4}}}{\bf{,}}...\)are generated in the same way. Generally,\({\bf{j = 1,2,}}...{\bf{,}}\)after the value\({{\bf{X}}_{\bf{j}}}{\bf{ = }}{{\bf{x}}_{\bf{j}}}\)has been observed,\({{\bf{X}}_{{\bf{j + 1}}}}\)is chosen from a uniform distribution on the interval\(\left( {{{\bf{x}}_{\bf{j}}}{\bf{,1}}} \right)\). Find the value of\({\bf{E}}\left( {{{\bf{X}}_{\bf{n}}}} \right)\).
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