Question

# For continuous random variables X and Y with joint probability density function f(x,y)=\begin{cases}xe^{-(x+xy)} & x>0\ and\ y>0\\0 & otherwise\end{cases} Are X and Y independent? Explain.

Random variables
For continuous random variables X and Y with joint probability density function
$$f(x,y)=\begin{cases}xe^{-(x+xy)} & x>0\ and\ y>0\\0 & otherwise\end{cases}$$
Are X and Y independent? Explain.

2021-06-01

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.