Is it possible to find dependence between X and Y from distribution of X-Y? Consider X,Y ~ Norm(0,1), I am exploring all the possible ways to find dependence. Correlation, Eye Balling from scatter plot are I think obvious methods.
I trying to think in a different direction just to have a better understanding. So, When Y is dependent on X what would X-Y distribution look like? From what I am aware of X-Y ~ Norm(0,2) when X and Y are independent.
Consider a distribution show in the graph below, I want to comment on dependence just from the graph 
If this is not possible how can joint probability tell be about dependence.
 A: Say $X$ and $Y$ are normally distributed with means $\mu_X, \mu_Y$ and variances $\sigma_X^2, \sigma_Y^2$. We're interested in $Z = X - Y$.
If $X$ and $Y$ are independent, $Z$ is normally distributed with mean $\mu_x - \mu_Y$ and variance $\sigma_X^2 + \sigma_Y^2$.
If $X$ and $Y$ are jointly normally distributed (with correlation $\rho$), $Z$ is normally distributed with mean $\mu_X - \mu_Y$ and variance $\sigma_X^2 + \sigma_Y^2 - 2 \rho \sigma_X \sigma_Y$. Therefore, it's possible in this case to recover the correlation of $X$ and $Y$ from the variances of $X$, $Y$, and $Z$:
$$\rho = \frac{\sigma_X^2 + \sigma_Y^2 - \sigma_Z^2}{2 \sigma_X \sigma_Y}$$
In the general case where $X$ and $Y$ have joint distribution $p_{XY}$, $Z$ is distributed as:
$$p_Z(z) = \int_{-\infty}^\infty p_{XY}(x, x-z) dx$$
If you're interested in general measures of dependence, it would be good to read about mutual information, which is computed from the joint distribution. One interpretation is that MI measures how much knowing the value of one variable reduces uncertainty about the other. Another interpretation is that MI measures the difference between the actual joint distribution and what it would be under independence (i.e. the product of the marginal distributions).
