I need to find the co-relation coefficient of the given joint density function of two random variables $X$ and $Y$:
$$ f(x,y) = \left\{ \begin{array}{ll} \frac{1}{210}(2x+y) & \quad 2<x<6, 0<y<5 \\ 0 & \quad otherwise \end{array} \right. $$
What I have done so far is:
a) $E[X]$: $$ \int_{2}^{6} x \frac{1}{210}(2x+y) dy $$
b) $E[X^2]$: $$ \int_{2}^{6} x^2 \frac{1}{210}(2x+y) dy $$
c) $E[Y]$ : $$ \int_{0}^{5} y \frac{1}{210}(2x+y) dy $$
d) $E[Y^2]$ : $$ \int_{0}^{5} y^2 \frac{1}{210}(2x+y) dy $$
e) $Var[X]$: $$ E[X^2]-(E[X])^2 $$
f) $Var[Y]$: $$ E[Y^2]-(E[Y])^2 $$
g) $E[XY]$ : $$ \int_{0}^{5}\int_{2}^{6} xy \frac{1}{210}(2x+y) dxdy$$
Is my method till here correct?
Steps (a)-(d) are incorrect. Recall that $$\mathbb{E}[x] = \int x\,f_X(x)\,dx$$ over the appropriate region. In your steps, you are integrating the product of $x$ and $f_{X,Y}(x,y)$ (i.e.\ the joint distribution of $X$ and $Y$). You should first find the marginal distribution of X from the joint distribution by integrating over the appropriate range of $y$: $$ f_X(x) = \int f_{X,Y}(x,y)\,dy $$ In this case, the integral would be
$$ f_X(x) = \int_0^5 \frac{1}{210} (2x + y)\,dy $$ (note that the resulting expression for $f_X(x)$ is of course only valid for $2 < x < 6$ as suggested by the original domain).
Once you have the marginal distribution of $X$, you can find its expectation and variance. You'll want to do a similar process for $Y$.