Start with PCA and multiple regression or Start with multiple regression and PCA

Correlation of Rolling Two Dice
If A is a random variable responsible for calculating the sum of two independent rolls of a die, and B is the result of calculating the value of first roll minus the value second roll, is is true that A and B have a $cov(A,B)\ne 0$? In other words, is it true that they are correlated?
I've come to the conclusion that they must be correlated because they are not independent, that is, the event of A can have an impact on event B, but I remain stuck due to the fact that causation does not necessarily imply correlation.
I know that independence −> uncorrelation, but that the opposite isn't true.

If the coefficients of a simple regression line, $B_0$ and $B_1$, are the same then why are the regression lines of $y$ on $x$ and $x$ on $y$ different given the condition $r^2<1$. I have tried all the manipulation and graphical analysis I can but can't seem to see why this is happening.

A similarity/metric learning method that takes in the form of $x^{T}Wy=z$, where $x$ and $y$ are real valued vectors. For example, two images.
Breaking it into a more familiar form:
$x^{T}Wy=\sum _{ij}{w}_{ij}{x}_i{y}_j=z$
This essentially looks very similar to polynomial regression with only interactions between features (without the polynomials). i.e.
$z=f_w(x)=\sum _i{w}_i{x}_i+\sum _i\sum _{j=i+1}{w}_{ij}{x}_i{x}_j$
I was curious to see if the optimization for the matrix $W$ is the same as doing optimization for multivariate linear/polynomial regression, since $x$ and $y$ are fixed, and the only variate is the weight matrix $W$?

Multiple regression problems (restricted regression, dummy variables)
Q1.
Model 1: $Y=X_1{\beta }_1+\epsilon$
Model 2: $Y=X_1{\beta }_1+X_2{\beta }_2+\epsilon$
(a) Suppose that Model 1 is true. If we estimates OLS estrimator $b_1$ for ${\beta }_1$ in Model 2, what will happen to the size and power properties of the test?
(b) Suppose that Model 2 is true. If we estimates OLS estrimator $b_1$ for ${\beta }_1$ in Model 1, what will happen to the size and power properties of the test?
-> Here is my guess.
(a) $b_1$ is unbiased, inefficient estimator. (I calculated it using formula for "inclusion of irrelevant variable" and $b_1=(X_1^{\prime }{M}_2{X}_1{)}^{-1}{X}_1^{\prime }{M}_{2}Y$ where $M_2$ is symmetric and idempotent matrix) Inefficient means that it has larger variance thus size increases and power increases too.
(b) $b_1$ is biased, efficient estimator. (I use formular for "exclusion of relevant variable" and $b_1=(X_1^{\prime }{X}_1{)}^{-1}{X}_1^{\prime }Y$) Um... I stuck here. What should I say using that information?
Q2.
Let $Q$ and $P$ be the quantity and price. Relation between them is different across reions of east, west, south and north, and as well, for different 4 seasons. Construct a model.
-> Actually, I don't know well about dummy variables. So any please solve this problem to help me.

Correlation: Concept to FormulaIn digital signal processing, we calculate the correlation between two discrete signals by multiplying corresponding samples of the two signals and then adding the products. Where does this process/formula for correlation come from?
I understand the concept of correlation (similarity) between two signals. But I fail to understand how it translates to the formula that it does.
All the texts I have seen so far start with this formula and explain cross correlation, auto correlation, etc. None of them attempt to explain how the formula was derived in the first place.

regression $x(t)=at+b$, number of trials, and $R_2$ of the regression. How do I find the value and $95\mathrm{\%}$ confidence interval for the value of $V=x/t$?