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.

$X=$Lot & $Y=$Cost
Give a broken line linear model with a breakpoint at $250$:
$Y=B_0+B_1{X}_1+B_2{X}_2+B_3{X}_3+e$
where $X_2=0$ or $1$ depending on whether the lot size is $\ge 250$ or $<250$ and $X_3=X_1\cdot X_2.$.
Which hypothesis statement is equivalent to the statement: The two regression lines have the same intercept term?
$$\begin{gathered} H_0:B_0=0 \\ {H}_0:B_1=0 \\ {H}_0:B_2=0 \\ {H}_0:B_3=0 \end{gathered}$$
$B_0$ is obviously the intercept for simple linear regression models, however, the broken line phrasing is causing me to be confused on this. My initial instinct was to assume $B_0$ hypothesis was appropriate, but now I'm wondering if $B_2=0$ makes more sense.

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 $T=9X+3Y+2Z+2U+V+W$

Specific equations for the three parameters in a least-squares quadratic regression? I'm looking for something like ${\beta }_1=,{\beta }_2=,{\beta }_3=$ for each of $y={\beta }_1+{\beta }_{2}x+{\beta }_3{x}^2$. To be clear, the right side of each of these equations should be evaluateable, using the data, to find the parameter.

In logistic regression, the regression coefficients ($\widehat{{\beta }_0},\widehat{{\beta }_1}$) are calculated via the general method of maximum likelihood. For a simple logistic regression, the maximum likelihood function is given as
$\ell ({\beta }_0,{\beta }_1)=\prod _{i:y_i=1}p(x_i)\prod _{i^{\prime }:y_{i^{\prime }}=0}(1-p(x_{i^{\prime }})).$
What is the maximum likelihood function for $2$ predictors? Or $3$ predictors?

Let the joint distribution of (X, Y) be bivariate normal with mean vector $\left(\begin{array}{c}0\\ 0\end{array}\right)$ and variance-covariance matrix
$\left(\begin{array}{cc}1& 𝝆\\ 𝝆& 1\end{array}\right)$ , where $-𝟏<𝝆<𝟏$ . Let ${𝚽}_{𝝆}(𝟎,𝟎)=𝑷(𝑿\le 𝟎,𝒀\le 𝟎)$ . Then what will be Kendall’s $\tau$ coefficient between X and Y equal to?

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?

We have a situation where we have pairs of points where x=heights of fathers and y=heights of the sons of these fathers. The mean of $X$ is equal to the mean $Y$ and $SD(Y)=SD(X)$, and $0<R<1$. Now I am supposed to show that the expected height of a son whose father is shorter than avg. is also less than average, but by a smaller degree. First off I need help understanding the "regression effect" i.e. "Regression Towards the Mean." If the standard deviations are the same why is this happening?