I have some problem at how to determine if two random variables are independent or not. If X and Y are two continuous random variables and their join probability density function is listed below
$$ f(x,y) = \left\{\begin{matrix} \frac{1}{2}x^{3}e^{-xy-x}, x > 0, y > 0\\ 0, otherwise \end{matrix}\right. $$
My Question is that intuitively that random variable X is independent of Y since X won't affect the value of Y, but I calculated $f_{X}(x)$ and $f_{Y}(y)$ and found that $f(x,y) \neq f_{X}(x)f_{Y}(y)$
I want to why random variables X and Y are look independent and either value won't affect the value of the other random variable, but they are dependent actually?
Thank you!!!!!
Fixing e.g. $x=1$ we get $f(1,y)=\frac12e^{-y-1}$ for $y>0$ and $f(1,y)=0$ otherwise.
For some constant $c$ this function equalizes $c\cdot f_Y(y\mid X=1)$ where $f_Y(y\mid X=1)$ denotes the PDF of $Y$ under condition $X=1$.
If we do the same for $x=2$ we get $f(2,y)=8e^{-2y-2}$ for $y>0$ and $f(1,y)=0$ otherwise.
For some constant $d$ this function equalizes $d\cdot f_Y(y\mid X=2)$ where $f_Y(y\mid X=2)$ denotes the PDF of $Y$ under condition $X=2$.
Evidently the functions are of different shape which makes clear that $f_Y(y\mid X=1)$ and $f_Y(y\mid X=2)$ are definitely not the same.
Apparantly under condition $X=1$ the distribution of $Y$ differs from the distribution of $Y$ under condition $X=2$.
So the value that is taken by $X$ affects the distribution of $Y$.
This observation is enough to conclude that $Y$ and $X$ are not independent.