# Sample Correlation and Random Variables -- An Intuitive Explanation?

I'm mainly trying to understand what it means for Monte Carlo samples to be correlated. Can somebody fit this into the theory of what I know about random variables, covariance and correlation?

And then specifically, when it comes to Monte Carlo path tracing (within the path integral formulation), what would the paths be analogous to? Let's say I have two variables producing paths of length $l$. The way I understand it right now is that they produce samples from a $3l$-dimensional space -- 3 cartesian coordinates, one for each point on the path. Is this correct?

If I had two random variables generating paths of some length $l$, how would I measure the correlation?

• Can you give some more context for what kind of correlation you're asking about? To define correlation you need two variables (eg "correlation between $x$ and $y$"). So—Monte Carlo samples with correlation between what and what? Aug 7 '17 at 20:26
• For example in BDPT, I've read that there is path correlation between the different paths which are generated by connecting eye and light paths at different points, since the paths share vertices. Aug 7 '17 at 20:51
• I'm curious about this too. I think it's when you reuse the same random number for two different things, or derive a new "random number" from an old one. It starts to make patterns that using independent random numbers wouldn't have. I'd like a more precise / formal explanation though too. Aug 8 '17 at 17:51