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Chapter 5: Q9E (page 287)
Find the value of \(\left( {\begin{array}{*{20}{c}}{\frac{3}{2}}\\4\end{array}} \right)\).
The value of \(\left( {\begin{array}{{}{}}{\frac{3}{2}}\\4\end{array}} \right)\) is \(\frac{3}{{128}}\).
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Suppose that two random variables \({{\bf{X}}_{\bf{1}}}{\rm{ }}{\bf{and}}{\rm{ }}{{\bf{X}}_{\bf{2}}}\)have a bivariate normal distribution, and two other random variables \({{\bf{Y}}_{\bf{1}}}{\rm{ }}{\bf{and}}{\rm{ }}{{\bf{Y}}_{\bf{2}}}\)are defined as follows:
\(\begin{array}{*{20}{l}}{{{\bf{Y}}_{\bf{1}}}{\rm{ }} = {\rm{ }}{{\bf{a}}_{{\bf{11}}}}{{\bf{X}}_{\bf{1}}}{\rm{ }} + {\rm{ }}{{\bf{a}}_{{\bf{12}}}}{{\bf{X}}_{\bf{2}}}{\rm{ }} + {\rm{ }}{{\bf{b}}_{\bf{1}}},}\\{{{\bf{Y}}_{\bf{2}}}{\rm{ }} = {\rm{ }}{{\bf{a}}_{{\bf{21}}}}{{\bf{X}}_{\bf{1}}}{\rm{ }} + {\rm{ }}{{\bf{a}}_{{\bf{22}}}}{{\bf{X}}_{\bf{2}}}{\rm{ }} + {\rm{ }}{{\bf{b}}_{\bf{2}}},}\end{array}\)
where
\(\left| {\begin{array}{*{20}{l}}{{{\bf{a}}_{{\bf{11}}}}{\rm{ }}{{\bf{a}}_{{\bf{12}}}}}\\{{{\bf{a}}_{{\bf{21}}}}{\rm{ }}{{\bf{a}}_{{\bf{22}}}}}\end{array}} \right|\)
Show that \({{\bf{Y}}_{\bf{1}}}{\rm{ }}{\bf{and}}{\rm{ }}{{\bf{Y}}_{\bf{2}}}\) also have a bivariate normal distribution.
Suppose that events occur in accordance with a Poisson process at the rate of five events per hour.
a. Determine the distribution of the waiting time \({{\bf{T}}_{\bf{1}}}\) until the first event occurs.
b. Determine the distribution of the total waiting time \({{\bf{T}}_{\bf{k}}}\) until k events have occurred.
c. Determine the probability that none of the first k events will occur within 20 minutes of one another.
Suppose that a random sample of 16 observations is drawn from the normal distribution with a mean \({\bf{\mu }}\) and standard deviation of 12 and that independently another random sample of 25 observations is drawn from the normal distribution with the same mean \({\bf{\mu }}\) and standard deviation of 20. Let \(\overline {\bf{X}} \,\,{\bf{and}}\,\,\overline {\bf{Y}} \) denote the sample means of the two samples. Evaluate \({\bf{Pr}}\left( {\left| {\overline {\bf{x}} - \overline {\bf{y}} } \right|{\bf{ < 5}}} \right)\).
Suppose that seven balls are selected at random withoutreplacement from a box containing five red balls and ten blue balls. If \(\overline X \) denotes the proportion of red balls in the sample, what are the mean and the variance of \(\overline X \) ?
Evaluate the integral\(\int\limits_0^\infty {{e^{ - 3{x^2}}}dx} \)
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