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**BERRY** $\displaystyle Cov(X,Y) = E(XY) - E(X)E(Y)$

$\displaystyle \rho = \frac{Cov(X,Y)}{\sigma_X\sigma_Y}$

Figure out the expectation the joint distribution.

$\displaystyle E(XY) = \int_0^1\int_0^{1-x}xyf(x,y)dydx$

Figure out the conditional distributions of X and Y.

$\displaystyle f_X(x) = \int_0^{1-x}f(x,y)dy$

$\displaystyle f_Y(y) = \int_0^1 f(x,y)dx$

Figure out the expectation of X and Y.

$\displaystyle E(X) = \int_0^1 xf_X(x)dx$

$\displaystyle E(Y) = \int_0^{1-x} yf_Y(y)dy$

Figure out the variance of X and Y.

$\displaystyle \sigma_X^2 = E(X^2) - [E(X)]^2$

$\displaystyle \sigma_Y^2 = E(Y^2) - [E(X)]^2$

Plug in the appropriate functions and now you have correlation.