I am curious, when you have two Random Variables who have perfect correlation as in, $E[XY]^2 = E[X^2]*E[Y^2]$ what does this mean with regards to their relationship? I understand that in L^p spaces it would mean X is a scalar multiple of Y but how does this work with regards to probability theory? One could say their distribution functions are the same, maybe you can even get that the R.Vs are scalar multiples of each other but how does this effect sample values?
For example, suppose I have sampled X to be a certain value, does this say anything about what Y is?