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Chapter 5: Q6E (page 315)
If the m.g.f. of a random variable X is\(\psi \left( t \right) = {e^{{t^2}}}\,for - \infty < t < \infty \)What is the distribution of X?
Distribution of X is normal distribution with\(\mu = 0\)and\({\sigma ^2} = 2\)
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Suppose that \({X_1}\) and \({X_2}\) have a bivariate normal distribution for which \(E\left( {{X_1}|{X_2}} \right) = 3.7 - 0.15{X_2},\,\,E\left( {{X_2}|{X_1}} \right) = 0.4 - 0.6{X_1}\,\,and\,\,Var\left( {{X_2}|{X_1}} \right) = 3.64\)Find the mean and the variance of\({X_1}\) , the mean and the variance of \({X_2}\), and the correlation of \({X_1}\)and\({X_2}\).
Find the 0.25 and 0.75 quantiles of the Fahrenheit temperature at the location mentioned in Exercise 3.
Suppose that X and Y are independent random variables, X has the gamma distribution with parameters 伪1 and 尾, and Y has the gamma distribution with parameters 伪2 and 尾. Let U = X/ (X + Y) and V = X + Y. Show that (a) U has the beta distribution with parameters 伪1 and 伪2, and (b) U and V are independent.
Suppose that the two measurements from flea beetles in Example 5.10.2 have the bivariate normal distribution with\({{\bf{\mu }}_{{\bf{1}}{\rm{ }}}} = {\rm{ }}{\bf{201}},{{\bf{\mu }}_{\bf{2}}}{\rm{ }} = {\rm{ }}{\bf{118}},{{\bf{\sigma }}_{\bf{1}}}{\rm{ }} = {\rm{ }}{\bf{15}}.{\bf{2}},{{\bf{\sigma }}_{\bf{2}}}{\rm{ }} = {\rm{ }}{\bf{6}}.{\bf{6}},{\rm{ }}{\bf{and}}\,\,{\bf{\rho }} = {\rm{ }}{\bf{0}}.{\bf{64}}.\)Suppose that the same two measurements from a secondspecies also have the bivariate normal distribution with\({{\bf{\mu }}_{\bf{1}}}{\rm{ }} = {\rm{ }}{\bf{187}},{{\bf{\mu }}_{\bf{2}}}{\rm{ }} = {\rm{ }}{\bf{131}},{{\bf{\sigma }}_{\bf{1}}}{\rm{ }} = {\rm{ }}{\bf{15}}.{\bf{2}},{{\bf{\sigma }}_{\bf{2}}}{\rm{ }} = {\rm{ }}{\bf{6}}.{\bf{6}},{\rm{ }}{\bf{and}}\,\,{\bf{\rho }} = {\rm{ }}{\bf{0}}.{\bf{64}}\). Let\(\left( {{{\bf{X}}_{\bf{1}}},{\rm{ }}{{\bf{X}}_{\bf{2}}}} \right)\) be a pair of measurements on a flea beetle from one of these two species. Let \({{\bf{a}}_{\bf{1}}},{\rm{ }}{{\bf{a}}_{\bf{2}}}\)be constants.
a. For each of the two species, find the mean and standard deviation of \({{\bf{a}}_{\bf{1}}}{{\bf{X}}_{\bf{1}}}{\rm{ }} + {\rm{ }}{{\bf{a}}_{\bf{2}}}{{\bf{X}}_{\bf{2}}}.\)(Note that the variances for the two species will be the same. How do you know that?)
b. Find \({{\bf{a}}_{\bf{1}}},{\rm{ }}{{\bf{a}}_{\bf{2}}}\) to maximize the ratio of the difference between the two means found in part (a) to the standarddeviation found in part (a). There is a sense inwhich this linear combination \({{\bf{a}}_{\bf{1}}}{{\bf{X}}_{\bf{1}}}{\rm{ }} + {\rm{ }}{{\bf{a}}_{\bf{2}}}{{\bf{X}}_{\bf{2}}}.\)does the best job of distinguishing the two species among allpossible linear combinations.
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.
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