I was wondering if there was any significance to assuming $\mathbb{E}[X^2] = \mathbb{E}[X]^2$? If X is normally distributed, this would correpsonding to it having zero variance (which I guess makes it a dirac delta function, or something similar?)
But is there any other intepretation of this? In particular, if we have three random variables $X_1,X_2,X_3$ such that $X_1+X_2+X_3=1$, what assumption are we making if we say $\mathbb{E}[X_iX_j] = \mathbb{E}[X_i]\mathbb{E}[X_j]$. Is there any context in which doing this can make sense? If $i \neq j$, then that just means $X_i$ and $X_j$ are independent (at least for two of the three values of $i,j$, since as the sum is 1, they aren't really all independent). But can it make sense for $i=j$?
For a bit more context, I was thinking in the context of say a system of chemical reaction equations.
You can write down a forward equation that describes the time evolution of the probability distribution. You can then calculate expected values and end up with something that looks like
$$\frac{d\mathbb{E}[X_1]}{dt} = a\mathbb{E}[X_1]+b\mathbb{E}[X_2]+c\mathbb{E}[X_1X_3]+d\mathbb{E}[X_1X_2]$$
$$\frac{d\mathbb{E}[X_2]}{dt} = e\mathbb{E}[X_3]+f\mathbb{E}[X_1X_2]+g\mathbb{E}[X_2X_3]-h\mathbb{E}[X_1^2]$$
If instead you wrote down a deterministic system for the same system of reactions using the law of mass action, you would arrive at essentially the same system, but without the expected values. But that's comparable to $\mathbb{E}[X_1^2]$ being replaced by $\mathbb{E}[X_1]^2$. I was wondering if there was any way of thinking about the relationship between these two in terms variances/ covariances of the random variable?
