Problem 6
A drawer contains eight different pairs of socks. If six socks are taken at random and without replacement, compute the probability that there is at least one matching pair among these six socks. Hint: Compute the probability that there is not a matching pair.
Problem 6
Let the random variable \(X\) have mean \(\mu\), standard deviation \(\sigma\), and
\(\mathrm{mgf}\) \(M(t),-h
Problem 6
Let the probability set function of the random variable \(X\) be \(P_{X}(D)=\)
\(\int_{D} f(x) d x\), where \(f(x)=2 x / 9\), for \(x \in \mathcal{D}=\\{x:
0
Problem 6
For each of the following pdfs of \(X\), find \(P(|X|<1)\) and
\(P\left(X^{2}<9\right)\).
(a) \(f(x)=x^{2} / 18,-3
Problem 6
If the sample space is \(\mathcal{C}=\\{c:-\infty
Problem 6
Show that the following sequences of sets, \(\left\\{C_{k}\right\\}\), are nondecreasing, (1.2.16), then find \(\lim _{k \rightarrow \infty} C_{k}\). (a) \(C_{k}=\\{x: 1 / k \leq x \leq 3-1 / k\\}, k=1,2,3, \ldots\). (b) \(C_{k}=\left\\{(x, y): 1 / k \leq x^{2}+y^{2} \leq 4-1 / k\right\\}, k=1,2,3, \ldots\)
Problem 7
Show that the following sequences of sets, \(\left\\{C_{k}\right\\}\), are
nonincreasing, \((1.2 .17)\), then find \(\lim _{k \rightarrow \infty} C_{k}\).
(a) \(C_{k}=\\{x: 2-1 / k
Problem 7
If \(C_{1}\) and \(C_{2}\) are subsets of the sample space \(\mathcal{C}\), show that $$ P\left(C_{1} \cap C_{2}\right) \leq P\left(C_{1}\right) \leq P\left(C_{1} \cup C_{2}\right) \leq P\left(C_{1}\right)+P\left(C_{2}\right) $$
Problem 7
A pair of dice is cast until either the sum of seven or eight appears. (a) Show that the probability of a seven before an eight is \(6 / 11\). (b) Next, this pair of dice is cast until a seven appears twice or until each of a six and eight has appeared at least once. Show that the probability of the six and eight occurring before two sevens is \(0.546\).
Problem 7
Let \(X\) have a pmf \(p(x)=\frac{1}{3}, x=1,2,3\), zero elsewhere. Find the pmf of \(Y=2 X+1\)
