Direction of correlation There are two interdependent variables - experience and fun. Using a correlation, we can only say how fun and experience are related, and the strength of the relationship. My question is how do I check if (i) fun is influencing experience or (ii) experience is affecting fun? I am not clear how should the directional arrow be indicated, using the Pearson Correlation.
 A: I believe that a Pearson correlation is not capable of telling you which 'direction' the relationship is.
A Pearson correlation is essentially looking for the tendency of one variable to vary in a certain way with another variable, but can not establish a causal link in that way.
For example, it is possible that having a lot of fun opens someone up to a lot of new experiences (so fun is influencing experience). It is also possible that life experience allows one to enjoy more things as you discover hobbies (so experience is influencing fun). Even a mixture of the two is plausible, or a third variable is in play (maybe rich people have more experience and more fun, but fun and experience are not related to each other at all causally).
I think typically, without resorting to making several probably fairly complicated assumptions about how you think the variables interact, establishing causality would require either an experimental design (or if this is not possible, a quasi-experimental design). A correlational study can't give you this type of information, which is pretty much why we have to run a lot of experiments rather than just mining for correlations everywhere.
A: The short answer is that you can't use Pearson correlation to determine causality. This is directly related to the popular saying that causation means correlation, but not vise versa. 
To give a reason, I think it's best to look at Wilk's lambda as a test of significance. Briefly, Wilk's lambda can be used as a significance test for the correlation between two multivariate random vectors. Let's say you have two random vectors $\bf X$ and $\bf Y$, each containing features that quantify "fun" and "experience", respectively. The question is whether we can regress $\bf X$ given $\bf Y$. To address this, there are several multivariate tests that can be used that measure the "association" or relationship between the two random vectors $\bf X$ and $\bf Y$ (Kshisargar 1972). Wilk's test (which is known as Wilk's lambda) is proportional to 
$\Lambda \propto \prod_{i=1}^p (1-r_i^2) $
is fully determined by the correlation coefficients between $\bf X$ and $\bf Y$, where $r_i$ are the sample correlation coefficients between the vectors $\bf X$ and $\bf Y$. The sample correlation coefficients can be calculated using canonical correlation analysis given an equal number of observations from $\bf X$ and $\bf Y$. The distribution of $\Lambda$ determines how likely it is that $\bf Y$ can be used to regress $\bf X$. Alternatively, had we selected to use $\bf X$ as the regressor, $\Lambda$ would have been the same, since the sample correlation coefficients (and the correlation coefficients) are symmetric with respect to $\bf X$ and $\bf Y$. Therefore, however certain you are that $\bf X$ can cause $\bf Y$ (as far as correlation is concerned), you're equally certain that $\bf Y$ can cause $\bf X$. 
A: You always have to remember that correlation is not causality! No correlation coefficient will tell you the causal relationship of your variables. The only way you can identify causation might be to run linear regression with lagged variables (if you have time series data) and see what seems to predict the other based on time. Otherwise only logical reasoning will help.
https://en.wikipedia.org/wiki/Correlation_does_not_imply_causation
