Understanding Inferential Statistics through Examples

Is it a "standard" Math/Numerical-Analysis hack to add a relatively small number e.g. 1*10E-5 to the diagonal of a squared matrix to ensure LU Decomposition (or whichever decomposition algorithm is applicable)? As opposed to "partially/totally pivoting"?

What is degree of freedom in statistics?

How can we fit a set of data points to a hyperbola, a square root function or a logarithmic function?

Show that if $x_{t+l}=A{x}_t$ , then $\mathrm{\rho <!--\; \rho \; -->}(l)=1$ if $A>0$, and $\mathrm{\rho <!--\; \rho \; -->}(l)=-1$ if $A<0$.

How do you write an exponential function to model the situation. Then estimate the value of the function after 5 years. A population of 290 animals that increases at an annual rate of 9%?

What is the meaning of :$:s$ after a mathematical symbol?

Correlation matrix with same pairwise correlation coefficientQuestion
Correlation matrix. Consider 𝑛 random variables with the same pairwise correlation coefficient ${\mathrm{\rho <!--\; \rho \; -->}}_n$. Find the highest possible value of ${\mathrm{\rho <!--\; \rho \; -->}}_n$for
a) n=3
b) n=4
c) general, n $\ge <!--\; \ge \; -->$2
HINT: Correlation matrix must be positive semi-definite.
My Workings
This is what I infer from "same pairwise coefficients":
$\left(\begin{array}{ccc}1& {\mathrm{\rho <!--\; \rho \; -->}}_n& {\mathrm{\rho <!--\; \rho \; -->}}_n\\ {\mathrm{\rho <!--\; \rho \; -->}}_n& 1& {\mathrm{\rho <!--\; \rho \; -->}}_n\\ {\mathrm{\rho <!--\; \rho \; -->}}_n& {\mathrm{\rho <!--\; \rho \; -->}}_n& 1\end{array}\right)$
Because a correlation matrix is positive semi-definite, all principal minors have to be positive
For n=3, the principal minors calculations yield
$\mid H_1\mid$=1
$\mid H_2\mid$=1-${\mathrm{\rho <!--\; \rho \; -->}}_n^2$ $\ge <!--\; \ge \; -->0$ $\Rightarrow <!--\; \Rightarrow \; -->$${\mathrm{\rho <!--\; \rho \; -->}}_n$ $\le <!--\; \le \; -->1$
$\mid H_3\mid ={\mathrm{\rho <!--\; \rho \; -->}}_n^2$ - ${\mathrm{\rho <!--\; \rho \; -->}}_n$$\ge <!--\; \ge \; -->0$ $\Rightarrow <!--\; \Rightarrow \; -->$ ${\mathrm{\rho <!--\; \rho \; -->}}_n$ $\ge <!--\; \ge \; -->1$
$\mid H_4\mid =(1^3$+2${\mathrm{\rho <!--\; \rho \; -->}}_n^3$)-(3${\mathrm{\rho <!--\; \rho \; -->}}_n^2$)$\ge <!--\; \ge \; -->0$ . I solved for the roots and found 1
Conclusion: For n=3, max ${\mathrm{\rho <!--\; \rho \; -->}}_n$=1
I did the same method for n=4 and found max ${\mathrm{\rho <!--\; \rho \; -->}}_n$=1 again
My Problem
Result looks too simple and false. Method is tedious
I have no idea how to do the general case. (by induction ?)
Thank you for your help
INDUCTION ATTEMPT
$H_0$
$\left(\begin{array}{cc}1& {\mathrm{\rho <!--\; \rho \; -->}}_2\\ {\mathrm{\rho <!--\; \rho \; -->}}_2& 1\end{array}\right)$
$\Rightarrow <!--\; \Rightarrow \; -->-<!--\; -\; -->\frac{1}{2-<!--\; -\; -->1}\le <!--\; \le \; -->{\mathrm{\rho <!--\; \rho \; -->}}_2\le <!--\; \le \; -->1$
$H_n$: Pn: Suppose that for a nxn correlation matrix An with same pairwise coefficients ${\mathrm{\rho <!--\; \rho \; -->}}_n$, $-<!--\; -\; -->\frac{1}{n-<!--\; -\; -->1}\le <!--\; \le \; -->{\mathrm{\rho <!--\; \rho \; -->}}_n\le <!--\; \le \; -->1$ holds
$H_{n+1}$
$-<!--\; -\; -->\frac{1}{n-<!--\; -\; -->1}\le <!--\; \le \; -->{\mathrm{\rho <!--\; \rho \; -->}}_n\le <!--\; \le \; -->1$
$\Leftarrow <!--\; \Leftarrow \; -->\Rightarrow <!--\; \Rightarrow \; -->$$-<!--\; -\; -->\frac{1}{(n+1)-<!--\; -\; -->1}\le <!--\; \le \; -->{\mathrm{\rho <!--\; \rho \; -->}}_{n+1}\le <!--\; \le \; -->1$
$\Leftarrow <!--\; \Leftarrow \; -->\Rightarrow <!--\; \Rightarrow \; -->$$-<!--\; -\; -->\frac{1}{n}\le <!--\; \le \; -->{\mathrm{\rho <!--\; \rho \; -->}}_{n+1}\le <!--\; \le \; -->1$
And because for all $-<!--\; -\; -->\frac{1}{n}\ge <!--\; \ge \; -->-<!--\; -\; -->\frac{1}{n-<!--\; -\; -->1}$
$-<!--\; -\; -->\frac{1}{n-<!--\; -\; -->1}\le <!--\; \le \; -->{\mathrm{\rho <!--\; \rho \; -->}}_{n+1}\le <!--\; \le \; -->1$

Whether there is some connection between fitting probability distribution on some data set and linear regression? Or this two tools are for different problems?

Use the difference quotient to estimate the instantaneous rate of change in $f(x)=x^2$ at x=3?

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

Explain what does it mean this denotation: $$\underset{w}{min}||Xw-<!--\; -\; -->y|{|}_2^2$$

Correlation Function
Let X,Y be random variables. If $\mathrm{\rho <!--\; \rho \; -->}(X,Y)=a$ (Correlation), where $a\in <!--\; \in \; -->(0,1)$, what can be said about the relationship between X and Y? Is it true that Y=bX+c+Z, where Z is a random variable? If it is true then how is $|Z|$ related to the correlation a?

What are the prerequisites for regression analysis?

Why is the standardized regression coefficient in a regression model with more than one independent variable not the same as the correlation coefficient between x we interested in and y in a regression model with more than one independent variable?
$$\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_i}=\mathrm{c}\mathrm{o}\mathrm{r}(Y_i,X_i)\cdot <!--\; \cdot \; -->\frac{\mathrm{S}\mathrm{D}(Y_i)}{\mathrm{S}\mathrm{D}(X_i)}$$
So
$$\mathrm{c}\mathrm{o}\mathrm{r}(Y_i,X_i)=\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_i}\cdot <!--\; \cdot \; -->\frac{\mathrm{S}\mathrm{D}(X_i)}{\mathrm{S}\mathrm{D}(Y_i)}$$
The formula for the standardized regression coefficient is also:
$$standardizedBeta=\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_i}\cdot <!--\; \cdot \; -->\frac{\mathrm{S}\mathrm{D}(X_i)}{\mathrm{S}\mathrm{D}(Y_i)}$$
So shouldn't it be
$$standardizedBeta=\mathrm{c}\mathrm{o}\mathrm{r}(Y_i,X_i)$$?

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