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Chapter 1: Q2E (page 45)
Suppose that 18 red beads, 12 yellow beads, eight blue beads, and 12 black beads are to be strung in a row. How many different arrangements of the colors can be formed?
The colors can be formed in \(5.134987617 \times {10^{26}}\) no. of arrangements.
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Let \({A_1},{A_2}, \ldots \) be an arbitrary infinite sequence of events, and let \({B_1},{B_2}, \ldots \)be another infinite sequence of events defined as follows: \({B_1} = {A_1}\), \({B_2} = {A_1}^c \cap {A_2}\), \({B_3} = {A_1}^c \cap {A_2}^c \cap {A_3}\), \({B_4} = {A_1}^c \cap {A_2}^c \cap {A_3}^c \cap {A_4}\),鈥rove that
\(\Pr \left( {\bigcup\limits_{i = 1}^n {{A_i}} } \right) = \sum\limits_{i = 1}^n {\Pr \left( {{B_i}} \right)} \)for \(n = 1,2,3, \ldots \)
and that
\(\Pr \left( {\bigcup\limits_{i = 1}^\infty {{A_i}} } \right) = \sum\limits_{i = 1}^\infty {\Pr \left( {{B_i}} \right)} \)
Let \({{\bf{X}}_{\bf{1}}}{\bf{,}}{{\bf{X}}_{{\bf{2,}}...}}\)be a sequence of i.i.d. random variables having the normal distribution with mean 渭 and variance\({\sigma ^2}\). Let \({{\bf{\bar X}}_{\bf{n}}}{\bf{ = }}\frac{{\bf{1}}}{{\bf{n}}}\sum\limits_{{\bf{i = 1}}}^{\bf{n}} {{{\bf{X}}_{\bf{i}}}} \)be the sample mean of the first n random variables in the sequence. Show that \({\bf{P}}\left( {\left| {{{\bf{X}}_{\bf{n}}}{\bf{ - \mu }}} \right| \le {\bf{c}}} \right)\) converges to 1 as n 鈫� 鈭�. Hint: Write the probability in terms of the standard normal c.d.f. and use what you know about this c.d.f.
Suppose that the heights of the individuals in a certain population have a normal distribution for which the value of the mean 胃 is unknown and the standard deviation is 2 inches. Suppose also that the prior distribution of 胃 is a normal distribution for which the mean is 68 inches and the standard deviation is 1 inch. Suppose finally that 10 people are selected at random from the population, and their average height is found to be 69.5 inches.
a. If the squared error loss function is used, what is the Bayes estimate of 胃?
b. If the absolute error loss function is used, what is the Bayes estimate of 胃? (See Exercise 7 of Sec. 7.3).
Sketch the p.d.f. of the gamma distribution for each of the following pairs of values of the parameters \(\alpha \) and \(\beta \): (a)\(\alpha = \frac{1}{2}\) and \(\beta = 1\), (b) \(\alpha = 1\) and \(\beta = 1\), (c) \(\alpha = 2\) and \(\beta = 1\).
A student selected from a class will be either a boy or a girl. If the probability that a boy will be selected is 0.3, what is the probability that a girl will be selected?
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