Let's say that we have 2 discrete random variables, $X$ and $Y$ and we care about their joint PMF. If this joint PMF is as follows,
$$p_{X, Y} (x_1, y_1) = 2/9, \hspace{0.5cm} p_{X, Y} (x_2, y_1) = 4/9$$ $$p_{X, Y} (x_1, y_2) = 1/9, \hspace{0.5cm} p_{X, Y} (x_2, y_2) = 2/9$$
how can we know intuitively that $X$ and $Y$ are independent? By taking the definition it is easily proven, but how can I understand it without the definition?
I can see that, no matter the value $Y$ takes $(y_1$ or $y_2)$, the value of the PMF for $X=x_2$ is double the value of its value for $X=x_1$. But why does this make these 2 random variables independent??
Information about $Y$ still gives you info about $X$. If $Y=y_2$, $P(X=x_1)= 1/9$ but if $Y=y_1$, then we have that $P(X=x_1)=2/9$. So the probabilities change depending on the value of $Y$.
I probably didnt express my question perfectly because I am quite confused, so sorry about that. If someone could help me clear up the misunderstanding, I'd really appreciate it.
Thanks a lot.
"I can see that no matter the value $Y$ takes ($y_1$ or $y_2$), the value of the PMF for $X=x_2$ is double the value of its value for $X=x_1$. But why does this make these 2 random variables independent??"
Answer Regardless of the value of $Y$, $x_2$ is twice as likely as $x_1$, so in every option the conditional probability to have $X=x_2$ is $2/3$ and $X=x_1$ is $1/3$. Information about $Y$ does not change these probabilities, so they are independent.