Understanding Inferential Statistics through Examples

Consider a graph having $n=8$
vertices labeled $1,2,...,8$ . Suppose that each edge is independently present with probability p. The degree of vertex i, designated as $D_i$ , is the number of edges that have vertex i as one of its vertices. Find $Corr(D_i,D_j)$ , the correlation between $D_i$ and $D_j$ .

Correlation and causation data
What are some real life examples (data) in which high correlation:
1) implies causation
2) doesn't imply causation.
I know that there is a lot of data out there of weird correlations, like divorce rate and consuption of margarine, but the problem with this data is, that we don't really know if one is caused by the other, because nobody tested it, so we cannot strictly say that they're unrelated.

Causation doesn't imply correlation and correlation does not imply
causation
Let P := |corr(X,Y) > .5| let Q := exits a relation F: $X⟹<!--\; ⟹\; -->Y$
Then the often stated line of correlation does not imply causation is simply! $Q⟹<!--\; ⟹\; -->P$.
It is also true that causation does not imply correlation. So! $Q⟹<!--\; ⟹\; -->P$
But $(P⟹<!--\; ⟹\; -->Q)\vee <!--\; \vee \; -->(Q⟹<!--\; ⟹\; -->P)$ is a tautology.

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 $cov(A,B)\ne <!--\; \ne \; -->0$? 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.

a. Two variables have a positive linear correlation. Does the dependent variable increase or decrease as the independent variable increases? Explain.
b. Two variables have a negative linear correlation. Does the dependent variable increase or decrease as the independent variable increases?
c. Describe the range of values for the correlation coefficient, r.
d. What does the sample correlation coefficient r measure? Which value indicates a stronger correlation:
$r_1=0.975$ or $r_2=-<!--\; -\; -->0.987$. Explain, please.

1. In your own words, describe the difference between continuous and discrete data sets.
2. What is the difference between the independent and dependent variable? How do you determine which goes on which axis on your graph?
3. What are the four types of correlation?
4. How do they differ from one another?
5. In your own words, explain the difference between correlation and causation.

Bryan estimates the cost of a vacation to be $730. The actual cost of the vacation is $850. What is the percent error?

Independent random variables $X,Y,X,U,V,W$ have variance equal to 1. Find $\rho (S,T)$ - the correlation coefficient of random variables $S=3X+3Y+2Z+U+V+W$ and $T=9X+3Y+2Z+2U+V+W$

Contribution of each variable in multiple linear regression
What will be the best measure of the contribution of a variable in multiple linear regression? I was thinking of using the coefficient ratio as a marker of a variable's contribution.
For example:
If the equation is
$Y_{predicted}=a_1{X}_1+a_2{X}_2+a_3$
Then $X_1$'s contribution can be written down as:
$\frac{a_1}{a_1+a_2+a_3}$
Is there some other method possible to write down the contribution. Since in this case if any coefficient is negative, there is a possibility that a variable's contribution exceeds $100\mathrm{\%<!--\; \%\; -->}$

A consumer organization estimates that over a 1-year period 20% of cars will need to be repaired once, 6% will need repairs twice, and 2% will require three or more repairs. What is the probability that a car chosen at random will need repairs?

What is a standardized regression coefficient?

Let $Y_1$ and $Y_2$ be independent random variables with $Y_1\sim N(1,3)$ and $Y_2\sim N(2,5)$ . If $W_1=Y_1+2Y_2$ and $W_2=4Y_1-Y_2$ , what is the joint distribution of $W_1$ and $W_2$ ?

What do we mean when we say that correlation does not imply causality? What are some of the ways in which an empirical analyst attempts to disentangle the two?
Provide examples.

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.

Multiple regression problems (restricted regression, dummy variables)
Q1.
Model 1: $Y=X_1{\mathrm{\beta <!--\; \beta \; -->}}_1+\mathrm{\epsilon <!--\; \epsilon \; -->}$
Model 2: $Y=X_1{\mathrm{\beta <!--\; \beta \; -->}}_1+X_2{\mathrm{\beta <!--\; \beta \; -->}}_2+\mathrm{\epsilon <!--\; \epsilon \; -->}$
(a) Suppose that Model 1 is true. If we estimates OLS estrimator $b_1$ for ${\mathrm{\beta <!--\; \beta \; -->}}_1$ 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 ${\mathrm{\beta <!--\; \beta \; -->}}_1$ 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.

Given the linear correlation coefficient r and the sample size n, determine the critical values of r and use your finding to state whether or not the given r represents a significant linear correlation. Use a significance level of 0.05. $r=0.767,n=25$

a. Critical values: $r=\pm <!--\; \pm \; -->0.396$, no significant linear correlation
b. Critical values: $r=\pm <!--\; \pm \; -->0.487$ , no significant linear correlation
c. Critical values: $r=\pm <!--\; \pm \; -->0.396$ , significant linear correlation
d. Critical values: $r=\pm <!--\; \pm \; -->0.487$ , significant linear correlation

The process to generate the two RVs is as follows. We first draw T from $Uniform(0,1)$ . If $T\le <!--\; \le \; -->0.5$ we take $X=T$ and draw Y from $Uniform(0,1)$ . Otherwise if $T>0.5$ , we take $Y=T$ and draw X from $Uniform(0,1)$ . Running a simulation it seems like X and Y are positively correlated, though intuitively it seems like they should have no effect on each other. What is the explanation?

Suppose $Z(t)={\Sigma }_{k=1}^{n}X{e}^{j({𝜔}_{0}t+{𝚽}_k)}$, $t\in R$ where ${𝜔}_0$ is a constant, n is a fixed positive integer, $X_1,...,X_n, {𝚽}_1,...,{𝚽}_n$ are mutually independent random variables, and $E{X}_k=0,D{X}_k={\sigma }_k^2,𝚽$ , $U[0,2\pi ],k=1,2,...,n$ . Find the mean function and correlation function of $\{Z(t),t\in <!--\; \in \; -->R\}$ .
I have tried to solve it.
For mean function,
$m_Z(s)=E\{Z_s\}=E\{X_s\}+iE\{Y_t\}$
$=E\{{\Sigma }_{k=1}^{s}X{e}^{j({𝜔}_{0}t+{𝚽}_k)}\}$
For correlation function,
$R_Z(s,u)=E\{Z_s,Z_u\}$
$=E\{Y(s)Y(u)\}$
$=E\{{\Sigma }_{k=1}^{s}X{e}^{j({𝜔}_{0}t+{𝚽}_k)}{\Sigma }_{k=1}^{u}X{e}^{j({𝜔}_{0}t+{𝚽}_k)}\}$
$=E\{{\Sigma }_{k=1}^s{\Sigma }_{k=1}^{u}X{e}^{j({𝜔}_{0}t+{𝚽}_k)}X{e}^{j({𝜔}_{0}t+{𝚽}_k)}\}$
I am stuck here. How to move from here ahead?

Start with PCA and multiple regression or Start with multiple regression and PCA
I would like to know something easy but very important.
Imagine I have a database with 0 NA, a perfect database who has been clean. And I have to do a PCA on this database. This datebase got a lot of individuals and variables ( 95 individuals and 10 variables)
I have to do a multiple regression and a PCA.
I must start per my multiple regression and eventually delete somme individuals who has been a Cook's distance > at the limit. And after I do my PCA on " new data base"
OR I must start per my PCA on my complete database, and after I do my multiple regression.
In conclusion, I must do :
- PCA
- multiple Regression
or
-multiple Regression
-PCA
Ty for helping me !

PCA vs Correlation
What is the relationship between (first) principal component(s) and the correlation matrix or the average correlation of the data. For example, in an empirical application I observe that the average correlation is almost the same as the ratio of the variance of the first principal component (first eigenvalue) to the total variance (sum of all eigenvalues).
Is there a mathematical relationship?