What is the relationship between the correlation of two variables and their covariance?

Which of the following statements is not correct for the relation R defined by aRb, if and only if b lives within one kilometre from a?
A) R is reflexive
B) R is symmetric
C) R is not anti-symmetric
D) None of the above

A line segment is a part of a line as well as a ray. True or False

Which characteristic of a data set makes a linear regression model unreasonable?

Find the meaning of 'Sxx' and 'Sxy' in simple linear regression

In the least-squares regression line, the desired sum of the errors (residuals) should be
a) zero
b) positive
c) 1
d) negative
e) maximized

Can the original function be derived from its $k^{th}$ order Taylor polynomial?

Should the independent (or dependent) variables in a linear regression model be normal or just the residual?

What kind of technique is to be adopted if I have to find an equation or model for say, $D$ depends on $C$, $C$ changes for a set of $B$, which changes for different $A$.

Correlation bound
Let x and y be two random variables such that:
Corr(x,y) = b, where Corr(x,y) represents correlation between x and y, b is a scalar number in range of [-1, 1]. Let y' be an estimation of y. An example could be y'=y+(rand(0,1)-0.5)*.1, rand(0,1) gives random number between 0, 1. I am adding some noise to the data.
My questions are:
Is there a way where I can bound the correlation between x, y' i.e. Corr(x,y')?I mentioned y' in light of random perturbation, I would like to know what if I don't have that information, where I only know that y' is a estimation of y. Are there any literature that cover it?

What is the benefit of OU vs regression for modeling data, say data in the form of ($x,y$) pairs?

Can you determine the correlation coefficient from the coefficient of determination?

How can one find the root of sesquilinear form with positive definite matrix?

From numerical simulation and regression analysis I discovered that the root-mean-square amplitude of white noise with bandwidth $\mathrm{\Delta <!--\; \Delta \; -->}f$ is proportional to $\sqrt{\mathrm{\Delta <!--\; \Delta \; -->}f}$. How can this be derived mathematically ?

In a Simple Linear Regression analysis, independent variable is weekly income and dependent variable is weekly consumption expenditure. Here $95$% confidence interval of regression coefficient, ${\mathrm{\beta <!--\; \beta \; -->}}_1$ is $(.4268,.5914)$.

How to find AIC values for both models using $R$ software?