Use the following data to determine a linear regression equation. (-3,4.5), (-2,4), (-2,3), (-1,3), (-1,2), (0,0), (0,1), (2,-1)

Let

$$Y _ { 1 } , Y _ { 2 } , \ldots , Y _ { n }$$

denote a random sample from a normal population with known mean

$$\mu _ { o }$$

and unknown variance 1/v. In this case,

$$U = \sum \left( Y _ { i } - \mu _ { o } \right) ^ { 2 }$$

is a sufficient statistic for v, and

$$W = v U$$

has a

$$\chi ^ {2}$$

distribution with n degrees of freedom. Use the conjugate gamma

$$( \alpha , \beta )$$

prior for v to do the following. Show that the marginal density of U is

$$m ( u ) = \frac { u ^ { ( n / 2 ) - 1 } } { \Gamma ( \alpha ) \Gamma ( n / 2 ) \beta ^ { \alpha } 2 ^ { ( n / 2 ) } } \left( \frac { 2 \beta } { u \beta + 2 } \right) ^ { ( n / 2 ) + \alpha } \Gamma \left( \frac { n } { 2 } + \alpha \right)$$

.

Fill in the blank or answer true/false without referring back to the text. The corresponding eigenfunction for the boundary-value problem $y^{\prime \prime}+\lambda y=0, \quad y^{\prime}(0)=0, \quad y(\pi / 2)=0$ for $\lambda=25$ is _____.

The Gray family is putting in a pool in the shape of a rectangular prism. The first plan shows a pool that is 15 feet long, 12 feet wide, and 5 feet deep. The second plan shows a pool with the same length and width, but a depth of 7 feet. How much more water is needed to fill the second pool if both pools are filled to the top?

On a number line, point K is to the left of point H and to the right of point R. Point T is between points R and K. Order the points from least to greatest.

Prove that if F is a subfield of a field E, and $c \in E,$ then $\theta: F[x] \rightarrow E$ defined by $\theta(f(x))= f(c)$ is a homomorphism.

Indicate whether the following statements are true or false. Explain. When a closed system is at the dead state, it is in thermal and mechanical equilibrium with the exergy reference environment, and the values of the system's energy and thermomechanical exergy are each zero.

Each pair of values is from an inverse variation. Find the missing value. (x,12), (4,1.5)

Find an SVD of the indicated matrix.

$$A = \left[ \begin{array} { r r } { 0 } & { - 2 } \\ { - 3 } & { 0 } \end{array} \right]$$

Lana says there are $4 ^ { 3 }$ eggs in a dozen. Patrick says there are $3 ^ { 4 }$ eggs in a dozen. Is either person correct? Explain your reasoning.

Perform the indicated operations. Where possible, reduce the answer to its lowest terms.

$$\frac { 1 } { 3 } - 5 \left( \frac { 1 } { 2 } + 1 \right)$$

Write the fraction or mixed number as a decimal. 10 4/9

Fill in the blank. One _______ is the measure of a central angle that intercepts an arc equal in length to the radius of the circle.

Name the property of real numbers illustrated by each equation. $\frac{2}{5}+\frac{27}{5} \cdot \frac{5}{27}=\frac{2}{5}+1$