Problem on bivariate probability distribution

Question

Suppose the bivariate probability distribution of a random vector ( X,Y) is given according to the distribution in the following table

X

y 0 1 2

0 1/9 2/9 1/9

1 2/9 2/9 0

2 1/9 0 0

Show that random variables X AND Y are not independent and find P(X=0IY=2)P(X-1IY-2)P(X-2IY-2)

Answer 1

If x and y independent then P(X|Y) = P(X). That is not true as the probability of X changes with the value of Y.

P(x=0|y=2) = 1/9

P(x=1|y=2) = 0

P(x=2|y=2) = 0