In probability theory, $X$ and $Y$ are independent if: $f_{X|Y}(x|y)=f_X(x)f_Y(y)$
If I have sample $Y_1,...,Y_n$ and I would like to test if $Y_i$ is independent from the rest of the sample, I wonder if showing:
$ f_{Y_1|Y_2,...Y_n}(y_1|y_2,...,y_n)=f_{Y_1}(y_1)...f_{Y_n}(y_n) $,etc.., would be the right way to do or there will be another way to test for independence of random variable $Y$? ( I found it is hard to show the above condition if I dont know the distribution/density function
Thank you so much in advance for any recommendation.