Show that C(u, v) = uv is a copula. Why is it called “the independence copula”?

Test the Spearman rank correlation coefficient. The table shows the average hours worked per week and the numbers of on-the-job injuries for a random sample of U.S. construction companies in a recent year. At $\alpha=0.05$, can you conclude that there is a significant correlation between average hours worked and the number of on-the-job injuries?

$$\begin{matrix} \text{Hours worked} & \text{38} & \text{38} & \text{37} & \text{38} & \text{38} & \text{40} & \text{39} & \text{39} & \text{39} & \text{40} & \text{39} & \text{41}\\ \text{Injuries} & \text{11} & \text{11} & \text{9} & \text{10} & \text{10} & \text{17} & \text{15} & \text{14} & \text{14} & \text{16} & \text{15} & \text{17}\\ \text{Hours worked} & \text{41} & \text{42} & \text{41} & \text{41} & \text{41} & \text{42} & \text{42} & \text{42} & \text{42} & \text{41} & \text{41} & \text{39}\\ \text{Injuries} & \text{17} & \text{21} & \text{18} & \text{18} & \text{18} & \text{22} & \text{21} & \text{19} & \text{21} & \text{18} & \text{17} & \text{12}\\ \text{Hours worked} & \text{38} & \text{38} & \text{39} & \text{39} & \text{36} & \text{37} & \text{36} & \text{37} & \text{37} & \text{37} & \text{37} & \text{ }\\ \text{Injuries} & \text{12} & \text{11} & \text{13} & \text{12} & \text{6} & \text{6} & \text{6} & \text{6} & \text{7} & \text{8} & \text{7} & \text{ }\\ \end{matrix}$$

For the following exercise, for each pair of functions, find a. $(f \circ g)(x)$ and b. $(g \circ f)(x)$ Simplify the results. Find the domain of each of the results. $f(x)=\frac{3}{2 x+1}, g(x)=\frac{2}{x}$

Determine the number of outcomes in the event. Then decide whether the event is a simple event or not. Explain your reasoning. You randomly select one card from a standard deck of 52 playing cards. Event A is selecting a diamond.

Consider a Poisson process on the real line, and denote by $N\left(t_{1}, t_{2}\right)$ the number of events in the interval $\left(t_{1}, t_{2}\right)$. If $t_{0}<t_{1}<t_{2}$, find the conditional distribution of $N\left(t_{0}, t_{1}\right)$ given that $N\left(t_{0}, t_{2}\right)=n$. (Hint: Use the fact that the numbers of events in disjoint subsets are independent.)

Fill in the blanks. a. A function f has an absolute maximum at c if ____ for all x in the domains D of f. The number f(c) is called the ___ ___ ___ of f on D. b. A function f has a relative minimum at c if _____ for all values of x is some ____ ___ containing c.

A drawer of socks contains seven black socks, eight blue socks, and nine green socks. Two socks are chosen in the dark. a. What is the probability that they match? b. What is the probability that a black pair is chosen?

Find the Taylor series about the given point for each of the functions. In each case find the radius of convergence. $f(x)=\sin x^{2}$ about $x_{0}=0$

Prove that if K is a normal subgroup of G then $K^{\prime} \leq G$.

If Y has a binomial distribution with n trials and success probability p, show that Y/n is a consistent estimator of p.

Use the lattice of subgroups of $Q D_{16}$ to help find the normalizers (a) $N_{Q D_{16}}(\langle\tau \sigma\rangle)$ (b) $N_{Q D_{16}}\left(\left\langle\tau, \sigma^{4}\right\rangle\right)$

Define $T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ by $T(\mathbf{x})=A \mathbf{x}$. Find a basis $\mathcal{B}$ for $\mathbb{R}^{2}$ with the property that $[T]_{\mathcal{B}}$ is diagonal.

$$A=\left[\begin{array}{rr} 4 & -2 \\ -1 & 5 \end{array}\right]$$

(a) find the intervals where the function f is increasing and where it is decreasing, (b) find the relative extrema of f, (c) find the intervals where the graph of f is concave upward and where it is concave downward, and (d) find the inflection points, if any, of f. $f(x)=\frac{x^{2}}{x-1}$

The price of a certain commodity in dollars per unit at time t (measured in weeks) is given by $p=8+4 e^{-2 t}+t e^{-2 t}$. a. What is the price of the commodity at t=0? b. How fast is the price of the commodity changing at t=0? c. Find the equilibrium price of the commodity.