# Why are the random variables X and Y independent in this case?

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.

• The mistake in your reasoning is when you say that, with $Y=y_2$, $P(X=x_1)=1/9$. In fact, with $Y=y_2$, the probabilities of $X$ should be calculated as conditional probabilities. $P(X=x_1\mid Y=y_2)=1/3$ and so is $P(X=x_1\mid Y=y_1)=1/3$, so the marginal distribution of $X$ does not depend on $Y$. (In other words, knowing $Y$ does not reveal any information about $X$ that is not already known.)
– user700480
Sep 2, 2020 at 10:15
• Thanks, you are correct of course. So basically with think with respect to conditional probabilities, because thats where independence comes in. Independence of two events means that new information about one does not change my beliefs about the other, and that is expressed via conditional probabilities. Sep 2, 2020 at 12:24

"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.
• Thanks for the answer. So essentialy I have to think this with regards to conditional probabilities, because thats where independence comes in. Independence means that that NEW info does not change my beliefs, and that is expressed via conditional probabilities. In addition to this, the way you make the aforementioned calculations is this? : $P(X=x_1|Y=y_1)=\frac{P(X=x_1, Y=y_1)}{P(Y=y_1)}=\frac{2/9}{2/9+4/9}=2/6=1/3$, is that correct? And that is equal to $P(X=x_1|Y=y_2)=1/3$. Sep 2, 2020 at 12:20