Chapter 6: Q5E (page 358)
How large a random sample must be taken from a given distribution in order for the probability to be at least 0.99 that the sample mean will be within 2 standard deviations of the mean of the distribution?
25
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Suppose that X has the Poisson distribution with mean 10. Use the central limit theorem, both without and with the correction for continuity, to determine an approximate value for\({\rm P}\left( {8 \le X \le 12} \right)\)Use the table of Poisson probabilities given in the back of this book to assess the quality of these approximations.
Suppose that \({{\bf{X}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{X}}_{\bf{n}}}\) form a random sample from a normal distribution with unknown mean ฮธ and variance \({{\bf{\sigma }}^{\bf{2}}}\). Assuming that \({\bf{\theta }} \ne {\bf{0}}\) , determine the asymptotic distribution of \({{\bf{\bar X}}_{\bf{n}}}^{\bf{3}}\)
Prove that if a sequence\({Z_1},{Z_2},...\)converges to a constant b in quadratic mean, then the sequence also converges to b in probability.
In this exercise, we construct an example of a sequence of random variables \({Z_n}\) such that but \(\Pr \left( {\mathop {\lim }\limits_{n \to \infty } \;{Z_n} = 0} \right) = 0\).That is, \({Z_n}\)converges in probability to 0, but \({Z_n}\)does not converge to 0 with probability 1. Indeed, \({Z_n}\)converges to 0 with probability 0.
Let Xbe a random variable having the uniform distribution on the interval\(\left[ {0,1} \right]\). We will construct a sequence of functions \({h_n}\left( x \right)\;for\;n = 1,2,...\)and define\({Z_n} = {h_n}\left( X \right)\). Each function \({h_n}\) will take only two values, 0 and 1. The set of x where \({h_n}\left( x \right) = 1\) is determined by dividing the interval \(\left[ {0,1} \right]\)into k non-overlapping subintervals of length\(\frac{1}{k}\;for\;k = 1,2,...\)arranging these intervals in sequence, and letting \({h_n}\left( x \right) = 1\) on the nth interval in the sequence for \(n = 1,2,...\)For each k, there are k non overlapping subintervals, so the number of subintervals with lengths \(1,\frac{1}{2},\frac{1}{3},...,\frac{1}{k}\) is \(1 + 2 + 3 + ... + k = \frac{{k\left( {k + 1} \right)}}{2}\)
The remainder of the construction is based on this formula. The first interval in the sequence has length 1, the next two have length ยฝ, the next three have length 1/3, etc.
Show that, for every \(x \in \left[ {0,1} \right),\;{h_n}\left( x \right) = 1\) for one and only one n among \(1 + \frac{{\left( {{k_n} - 1} \right){k_n}}}{2},...,\frac{{{k_n}\left( {{k_n} + 1} \right)}}{2}\)
Let X be a random variable for which \({\bf{E}}\left( {\bf{X}} \right){\bf{ = \mu }}\)and\({\bf{Var}}\left( {\bf{X}} \right){\bf{ = }}{{\bf{\sigma }}^{\bf{2}}}\).Construct a probability distribution for X such that \({\bf{P}}\left( {\left| {{\bf{X - \mu }}} \right| \ge {\bf{3\sigma }}} \right){\bf{ = }}\frac{{\bf{1}}}{{\bf{9}}}\)
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