Understanding Inferential Statistics through Examples

If f(3)=8 and f'(3)=-4, then how do you use linear approximation to estimate f(3.02)?

Let C be a positive definite correlation matrix partitioned as
$C=\left[\begin{array}{cc}{I}_{k_1}& A\\ A^{\prime }& I_{k_2}\end{array}\right]$
How can I show that the eigenvalues of AA′ are all less than 1?

How to, using Taylor series approximation , estimate the value of $\mathrm{\pi <!--\; \pi \; -->}$ , when $\arctan <!--\; \; -->(x)\approx <!--\; \approx \; -->x-<!--\; -\; -->x^3/3+x^5/5-<!--\; -\; -->x^7/7$?

variance of multiple regression coefficients
If I consider universal kriging (or multiple spatial regression) in matrix form as:
$\mathbf{V}\mathbf{=}\mathbf{X}\mathbf{A}\mathbf{+}\mathbf{R}$
where $\mathbf{R}$ is the residual and $\mathbf{A}$ are the trend coefficients, then the estimate of $\stackrel{\mathbf{^<!--\; ^\; -->}}{\mathbf{A}}$ is:
$\stackrel{\mathbf{^<!--\; ^\; -->}}{\mathbf{A}}=({\mathbf{X}}^{\mathbf{T}}{\mathbf{C}}^{\mathbf{-<!--\; -\; -->}\mathbf{1}}\mathbf{X}{\mathbf{)}}^{\mathbf{-<!--\; -\; -->}\mathbf{1}}{\mathbf{X}}^{\mathbf{T}}{\mathbf{C}}^{\mathbf{-<!--\; -\; -->}\mathbf{1}}\mathbf{V}$
(as I understand it), where $\mathbf{C}$ is the covariance matrix, if it is known. Then, the variance of the coefficients is:
VAR($\text{VAR}(\stackrel{\mathbf{^<!--\; ^\; -->}}{\mathbf{A}})=({\mathbf{X}}^{\mathbf{T}}{\mathbf{C}}^{\mathbf{-<!--\; -\; -->}\mathbf{1}}\mathbf{X}{\mathbf{)}}^{\mathbf{-<!--\; -\; -->}\mathbf{1}}$???
How does one get from the estimate of $\stackrel{\mathbf{^<!--\; ^\; -->}}{\mathbf{A}}$, to its variance? i.e. how can I derive that variance?

If we have two non-zero correlated random variables then they are dependent.
Why then do we have the saying "Correlation does not imply Causation". A change in one variable may not cause exactly the same change in another but there is at least some 'causal' link.

Multiple regression degrees of freedom $f$-test.
I'm finding conflicting information from college textbooks on calculating the degrees of freedom for a a global $F$-test on a multiple regression. To be absolutely clear, assume there are 50 observations and 3 independent variables. Can you please tell me the df for the numerator and denominator? I have found 2 sets of numbers in college texts. One indicating the numerator is equal to $P$, in this case 3, and alternatively $P-<!--\; -\; -->1$. For the denominator I am finding $n-<!--\; -\; -->p$,which in this case would be 47, and alternatively, $n-<!--\; -\; -->p-<!--\; -\; -->1$. Perhaps I am misunderstanding the material and there are circumstances when one vs. the other formula applies. I've not done any regression analysis in more than 25 years and now find I'm stuck on a Christmas vacation project I wanted to do with my son. So any help that would explain, in a gentle way, (I can't get through the quadratic explanation, or something that will bury me in calculus) how to determine the df would be appreciated. Concrete examples would be very beneficial. Also, if there is a good practical walk through of multiple regression/Anova that will show some examples and explain concepts (but please do not recommend Regression for Dummies) I'd appreciate a referral to that as well. Thanks for your help.

What’s Wrong?
Motorcycles and Sour Cream In recent years, there has been a strong correlation between per capita consumption of sour cream and the numbers of motorcycle riders killed in noncollision accidents. Therefore, consumption of sour cream causes motorcycle fatalities.

Multiple linear regression $b_0=0$
I am trying to calculate the coefficients $b_1,b_2,...$ of a multiple linear regression, with the condition that $b_0=0$. In Excel this can be done using the RGP Function and setting the constant to FALSE.
How can this be done with a simple Formular?
Thank you in Advance!

Advantages of Mathematics competition/olympiad students in Mathematical Research
Everyone in this community I think would be familiar with International Mathematical Olympiad, which is an International Mathematics Competition held for high school students, with many countries participating from around the world.
What's interesting to note is that many of the IMO participants have gone to win the Fields Medal. Notable personalities include Terence Tao (2006), Ngo Bao Chau (2010), Grigori Perelman (2006), etc.
I would like to know: What advantages does an IMO student possess over a 'normal' student in terms of mathematical research? Does the IMO competition help the student in becoming a good research mathematician or doesn't it?

Matrix derivative in multiple linear regression model
The basic setup in multiple linear regression model is
$\begin{array}{rl}Y& =\left[\begin{array}{c}{y}_1\\ y_2\\ ⋮<!--\; ⋮\; -->\\ y_n\end{array}\right]\end{array}$
$\begin{array}{rl}X& =\left[\begin{array}{cccc}1& x_{11}& \dots <!--\; \dots \; -->& x_{1k}\\ 1& x_{21}& \dots <!--\; \dots \; -->& x_{2k}\\ ⋮<!--\; ⋮\; -->& \dots <!--\; \dots \; -->& \dots <!--\; \dots \; -->\\ 1& x_{n1}& \dots <!--\; \dots \; -->& x_{nk}\end{array}\right]\end{array}$
$\begin{array}{rl}\mathrm{\beta <!--\; \beta \; -->}& =\left[\begin{array}{c}{\mathrm{\beta <!--\; \beta \; -->}}_0\\ {\mathrm{\beta <!--\; \beta \; -->}}_1\\ ⋮<!--\; ⋮\; -->\\ {\mathrm{\beta <!--\; \beta \; -->}}_k\end{array}\right]\end{array}$
$\begin{array}{rl}\mathrm{ϵ<!--\; ϵ\; -->}& =\left[\begin{array}{c}{\mathrm{ϵ<!--\; ϵ\; -->}}_1\\ {\mathrm{ϵ<!--\; ϵ\; -->}}_2\\ ⋮<!--\; ⋮\; -->\\ {\mathrm{ϵ<!--\; ϵ\; -->}}_n\end{array}\right]\end{array}$
The regression model is $Y=X\mathrm{\beta <!--\; \beta \; -->}+\mathrm{ϵ<!--\; ϵ\; -->}$
To find least square estimator of $\mathrm{\beta <!--\; \beta \; -->}$ vector, we need to minimize $S(\mathrm{\beta <!--\; \beta \; -->})={\mathrm{\Sigma <!--\; \Sigma \; -->}}_{i=1}^n{\mathrm{ϵ<!--\; ϵ\; -->}}_i^2={\mathrm{ϵ<!--\; ϵ\; -->}}^{\prime }\mathrm{ϵ<!--\; ϵ\; -->}=(y-<!--\; -\; -->x\mathrm{\beta <!--\; \beta \; -->}{)}^{\prime }(y-<!--\; -\; -->x\mathrm{\beta <!--\; \beta \; -->})=y^{\prime }y-<!--\; -\; -->2{\mathrm{\beta <!--\; \beta \; -->}}^{\prime }{x}^{\prime }y+{\mathrm{\beta <!--\; \beta \; -->}}^{\prime }{x}^{\prime }x\mathrm{\beta <!--\; \beta \; -->}$
$\frac{\mathrm{\partial <!--\; \partial \; -->}S(\mathrm{\beta <!--\; \beta \; -->})}{\mathrm{\partial <!--\; \partial \; -->}\mathrm{\beta <!--\; \beta \; -->}}=0$
My question: how to get $-<!--\; -\; -->2x^{\prime }y+2x^{\prime }x\mathrm{\beta <!--\; \beta \; -->}$?

Find the constrained least-squares estimator for a multiple regression model
Consider the multiple regression model
$Y=X\mathrm{\beta <!--\; \beta \; -->}+\mathrm{ϵ<!--\; ϵ\; -->}$
with the restriction that $\underset{l}{\overset{n}{\sum <!--\; \sum \; -->}}{b}_i=1$
I want to find the least squares estimator of $\mathrm{\beta <!--\; \beta \; -->}$, so I need to solve the following optimization problem
$min(Y-<!--\; -\; -->X\mathrm{\beta <!--\; \beta \; -->}{)}^t(Y-<!--\; -\; -->X\mathrm{\beta <!--\; \beta \; -->})$
$s.t.\underset{l}{\overset{n}{\sum <!--\; \sum \; -->}}{b}_i=1$
Let's set
$L=(Y-<!--\; -\; -->X\mathrm{\beta <!--\; \beta \; -->}{)}^t(Y-<!--\; -\; -->X\mathrm{\beta <!--\; \beta \; -->})-<!--\; -\; -->\mathrm{\lambda <!--\; \lambda \; -->}(U^t\mathrm{\beta <!--\; \beta \; -->}-<!--\; -\; -->1)=Y^{t}Y+{\mathrm{\beta <!--\; \beta \; -->}}^t{X}^{t}X\mathrm{\beta <!--\; \beta \; -->}+-<!--\; -\; -->2{\mathrm{\beta <!--\; \beta \; -->}}^t{X}^{t}Y-<!--\; -\; -->\mathrm{\lambda <!--\; \lambda \; -->}(U^t\mathrm{\beta <!--\; \beta \; -->}-<!--\; -\; -->1)$
where U is a dummy vector of ones (and therefore $U^T\mathrm{\beta <!--\; \beta \; -->}=\underset{l}{\overset{n}{\sum <!--\; \sum \; -->}}{b}_i$).
Take derivatives
$\frac{d}{d\mathrm{\beta <!--\; \beta \; -->}}=2X^{t}X\mathrm{\beta <!--\; \beta \; -->}-<!--\; -\; -->2X^{t}Y-<!--\; -\; -->\mathrm{\lambda <!--\; \lambda \; -->}{U}^t=0$
$\frac{d}{d\mathrm{\lambda <!--\; \lambda \; -->}}=U^t\mathrm{\beta <!--\; \beta \; -->}-<!--\; -\; -->1=0$
So from the first equation we can get an expression for $\mathrm{\beta <!--\; \beta \; -->}$, but what should I do with the $\mathrm{\lambda <!--\; \lambda \; -->}$? The second equation doesn't seem to be useful to get rid of it.

James types 84 words in 2 minutes. He estimates that his history essay is 420 words long. What is the least amount of time James will need to type his essay?

I need to determine multicollinearity of predictors, but I have only two. So, if VIF $\frac{1}{1-<!--\; -\; -->R_j^2}$ , then in case there are no other predictors VIF will always equal 1? So, maybe it's not even possible to gauge multicollinearity if there are only two predictors?

D.r Black is standing 20 feet from a streetlamp. The lamp is making his shadow 5 feet long. He estimates that the angle of elevation from the tip of his shadow to the top of the streetlamp is ${30}^{\circ <!--\; \circ \; -->}$. How tall is the street lamp to the nearest foot?

Let U and V be discrete random variables with $U=-(V+1)/2$ . What is $Corr(U,V)$ ?

Probability (independent?) events
Hi everyone I have a question about the following problem:
Events for a family: $A_1$ = ski, $A_2=$= does not ski, $B_1$ = has children but none in 8-16, $B_2$ = has some children in 8-16, and $B_3$ = has no children. Also, $P(A_1)=0.4$, $P(B_2)=0.35$, $P(B_1)=0.25$ and $P(A_1\cap <!--\; \cap \; -->B_1)=0.075$, $P(A_1\cap <!--\; \cap \; -->B_2)=0.245$. Find $P(A_1\cap <!--\; \cap \; -->B_1)=0.075$.
Here is my solution:
Since P(A1 and B1) = 0.075, P(A2 and B1) = 0.25-0.075= 0.175. Also since P(A1 and B2) = 0.245, P(A2 and B2) = 0.35 - 0.245 = 0.105. From this we can find P(A2 and B3) which is 0.6-0.175-0.105 = 0.32. But when I use the formula for independent events formula $P(A_2\cap <!--\; \cap \; -->B3)=P(A_2)\ast <!--\; \ast \; -->P(B3)$ I get 0.24. Does this mean that the events are not independent? If so, how are these events not independent?