The probability, \(P (X > 1\ and\ Y > 1)\) can be calculated using the joint probability density function.

If two random variables, X and Y are independent, then the joint density function can be written as a product of the marginal density function, that is,

\(f(x,y)=f_{X}(x)\cdot f_{Y}(y)\)

Here

\(f(x,y)=xe^{-(x+xy)}\)

\(f_{X}(x)=e^{-x}\)

\(f_{Y}(y)=\frac{1}{(y+1)^{2}}\)

\(f_{X}(x)\cdot f_{Y}(y)=\frac{e^{-x}}{(y+1)^{2}}\)

\(\neq xe^{-(x+xy)}\)

\(\Rightarrow f(x,y)\neq f_{X}(x)\cdot f_{Y}(y)\)

Thus, X and Y are not independent.