Let (X, Y ) be a bivariate random variable with the joint pdf

Show that X and Y are not independent but are uncorrelated.

$f_X(x)$ is : $$f_X(x) = \int\limits_{-\infty}^{\infty} f_{XY}(x,y) dy $$ $$ = \int\limits_{-\infty}^{\infty} \frac{x^2 + y^2}{4\Pi}e^{\frac{-(x^2+y^2)}{2}} dy $$ $$ =\frac{1}{4\Pi} \sqrt{2\Pi} (x^2+1) e^{\frac{-x^2}{2}} $$ $$ = \frac{1}{\sqrt{2\Pi}} (x^2+1) e^{\frac{-x^2}{2}} $$ $ f_Y(y)$ is :

$$ f_Y(y) = \int\limits_{-\infty}^{\infty} f_{XY}(x,y) dx $$ $$ =\int\limits_{-\infty}^{\infty} \frac{x^2 + y^2}{4\Pi}e^{\frac{-(x^2+y^2)}{2}} dx $$ $$ = \frac{1}{4\Pi} \sqrt{2\Pi} (y^2+1) e^{\frac{-y^2}{2}} $$ $$ = \frac{1}{\sqrt{2\Pi}} (y^2+1) e^{\frac{-y^2}{2}} $$

X and Y are dependent since $$\frac{x^2 + y^2}{4\Pi}e^{\frac{-(x^2+y^2)}{2}} \neq \frac{1}{8\Pi} e^{\frac{-(x^2+y^2)}{2}} (x^2+1) (y^2+1)$$

Correlation $$cov(x,y) = E[XY] - E[X] \cdot E[Y] $$

E[X] is :

$$E[X] = \int\limits_{-\infty}^{\infty} x f_X(x) dx $$ $$ = \int\limits_{-\infty}^{\infty} x \frac{1}{\sqrt{2\Pi}} (x^2+1) e^{\frac{-x^2}{2}} $$ $$ = 0 $$

E[Y] is : $$ E[Y] = \int\limits_{-\infty}^{\infty} y f_Y(y) dx $$ $$ = \int\limits_{-\infty}^{\infty} y \frac{1}{\sqrt{2\Pi}} (y^2+1) e^{\frac{-y^2}{2}} $$ $$ = 0 $$

E[XY] is : $$ E[XY] = \int\limits_{-\infty}^{\infty} xy f_{XY}(x,y) dxdy $$ $$ = \int\limits_{-\infty}^{\infty} xy \frac{x^2 + y^2}{4\Pi}e^{\frac{-(x^2+y^2)}{2}} dx dy $$ $$ = 0 $$

con(x,y) is : $$ cov(x,y) = E[XY] - E[X]E[Y] = 0 $$ $$ \rho = 0 $$

X and Y are not corelated