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Chapter 5: Q10E (page 316)
If a random sample of 25 observations is taken from the normal distribution with mean \(\mu \) and standard deviation 2, what is the probability that the sample mean will lie within one unit of 渭 ?
Probability that the sample mean will lie within one unit of 渭 is 0.9876.
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Suppose that the random variables \({{\bf{X}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{X}}_{\bf{k}}}\) are independent and that \({{\bf{X}}_{\bf{i}}}\) has the Poisson distribution with mean \({{\bf{\lambda }}_{\bf{i}}}\left( {{\bf{i = 1, \ldots ,k}}} \right)\). Show that for each fixed positive integer n, the conditional distribution of the random Vector \({\bf{X = }}\left( {{{\bf{X}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{X}}_{\bf{k}}}} \right)\), given that \(\sum\limits_{{\bf{i = 1}}}^{\bf{k}} {{{\bf{X}}_{\bf{i}}}{\bf{ = n}}} \) it is the multinomial distribution with parameters n and
\(\begin{array}{l}{\bf{p = }}\left( {{{\bf{p}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{p}}_{\bf{k}}}} \right){\bf{,}}\,{\bf{where}}\\{{\bf{p}}_{\bf{i}}}{\bf{ = }}\frac{{{{\bf{\lambda }}_{\bf{i}}}}}{{\sum\limits_{{\bf{j = 1}}}^{\bf{k}} {{{\bf{\lambda }}_{\bf{j}}}} }}\,{\bf{for}}\,{\bf{i = 1, \ldots ,k}}{\bf{.}}\end{array}\)
Suppose that the diameters of the bolts in a large box follow a normal distribution with a mean of 2 centimeters and a standard deviation of 0.03 centimeters. Also, suppose that the diameters of the holes in the nuts in another large box follow the normal distribution with a mean of 2.02 centimeters and a standard deviation of 0.04 centimeters. A bolt and a nut will fit together if the diameter of the hole in the nut is greater than the diameter of the bolt, and the difference between these diameters is not greater than 0.05 centimeter. If a bolt and a nut are selected at random, what is the probability that they will fit together?
Let,X1, X2andX3and be independent lifetimes of memory chips. Suppose that each\({X_i}\)has the normal distribution with mean 300 hours and standard deviation 10 hours. Compute the probability that at least one of the three chips lasts at least 290 hours.
Suppose that F is a continuous c.d.f. on the real line, and let \({{\bf{\alpha }}_{\bf{1}}}\,{\bf{and}}\,{{\bf{\alpha }}_{\bf{2}}}\)be numbers such that \({\bf{F}}\left( {{{\bf{\alpha }}_{\bf{1}}}} \right)\,{\bf{ = 0}}{\bf{.3}}\,{\bf{and}}\,{\bf{F}}\left( {\,{{\bf{\alpha }}_{\bf{2}}}} \right){\bf{ = 0}}{\bf{.8}}{\bf{.}}\). Suppose 25 observations are selected at random from the distribution for which the c.d.f. is F. What is the probability that six of the observed values will be less than \({{\bf{\alpha }}_{\bf{1}}}\), 10 of the observed values will be between \({{\bf{\alpha }}_{\bf{1}}}\) and \({{\bf{\alpha }}_{\bf{2}}}\), and nine of the observed values will be greater than \({{\bf{\alpha }}_{\bf{2}}}\)?
Suppose that\({X_1}and\,{X_2}\) have the bivariate normal distribution with means\({\mu _1}and\,{\mu _2}\) variances\({\sigma _1}^2and\,{\sigma _2}^2\), and correlation 蟻. Determine the distribution of\({X_1} - 3{X_2}\).
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