Problem 21
The distribution of the random variable \(X\) in Example \(1.7 .3\) is a member of
the log- \(F\) familily. Another member has the cdf
$$
F(x)=\left[1+\frac{2}{3} e^{-x}\right]^{-5 / 2}, \quad-\infty
Problem 22
Players \(A\) and \(B\) play a sequence of independent games. Player \(A\) throws a die first and wins on a "six." If he fails, \(B\) throws and wins on a "five" or "six." If he fails, \(A\) throws and wins on a "four," "five," or "six." And so on. Find the probability of each player winning the sequence.
Problem 22
Let \(X\) have the pdf \(f(x)=x^{2} / 9,0
Problem 22
Let \(X\) be a random variable of the discrete type with pmf \(p(x)\) that is positive on the nonnegative integers and is equal to zero elsewhere. Show that $$ E(X)=\sum_{x=0}^{\infty}[1-F(x)] $$ where \(F(x)\) is the cdf of \(X\).
Problem 22
Consider the events \(C_{1}, C_{2}, C_{3}\). (a) Suppose \(C_{1}, C_{2}, C_{3}\) are mutually exclusive events. If \(P\left(C_{i}\right)=p_{i}, i=1,2,3\), what is the restriction on the sum \(p_{1}+p_{2}+p_{3} ?\) (b) In the notation of part (a), if \(p_{1}=4 / 10, p_{2}=3 / 10\), and \(p_{3}=5 / 10\), are \(C_{1}, C_{2}, C_{3}\) mutually exclusive?
Problem 23
Let \(C_{1}, C_{2}, C_{3}\) be independent events with probabilities \(\frac{1}{2}, \frac{1}{3}, \frac{1}{4}\), respectively. Compute \(P\left(C_{1} \cup C_{2} \cup C_{3}\right)\).
Problem 23
Let \(X\) have the pmf \(p(x)=1 / k, x=1,2,3, \ldots, k\), zero elsewhere. Show that the \(\mathrm{mgf}\) is $$ M(t)=\left\\{\begin{array}{ll} \frac{e^{t}\left(1-e^{k t}\right)}{k\left(1-e^{t}\right)} & t \neq 0 \\ 1 & t=0 \end{array}\right. $$1.9.23. Let \(X\) have the pmf \(p(x)=1 / k, x=1,2,3, \ldots, k\), zero elsewhere. Show that the \(\mathrm{mgf}\) is $$ M(t)=\left\\{\begin{array}{ll} \frac{e^{t}\left(1-e^{k t}\right)}{k\left(1-e^{t}\right)} & t \neq 0 \\ 1 & t=0 \end{array}\right. $$
Problem 23
Suppose \(\mathcal{D}\) is a nonempty collection of subsets of \(\mathcal{C}\). Consider the collection of events $$ \mathcal{B}=\cap\\{\mathcal{E}: \mathcal{D} \subset \mathcal{E} \text { and } \mathcal{E} \text { is a } \sigma \text { -field }\\} $$ Note that \(\phi \in \mathcal{B}\) because it is in each \(\sigma\) -field, and, hence, in particular, it is in each \(\sigma\) -field \(\mathcal{E} \supset \mathcal{D}\). Continue in this way to show that \(\mathcal{B}\) is a \(\sigma\) -field.
Problem 23
If the pdf of \(X\) is \(f(x)=2 x e^{-x^{2}}, 0
Problem 24
From a bowl containing five red, three white, and seven blue chips, select four at random and without replacement. Compute the conditional probability of one red, zero white, and three blue chips, given that there are at least three blue chips in this sample of four chips.
