I asked a Large Language Model the following

Let X and Y be discrete random variables with the joint distribution given by P(X = 1; Y = -1) = 1/6 ; P(X = 2; Y = -1) = 1/6 ; P(X = 1; Y = 0) = 1/3 ; P(X = 2; Y = 0) = 1/6 ; P(X = 2; Y = 2) = 1/6 ; and P(X = x; Y = y) = 0; for (x; y) not in {(1,-1), (2,-1), (1, 0), (2, 0), (2, 2)}. Stating any module results you use, determine if X and Y are independent.

It came back with the following, can you spot where it changes the question?

To determine if X and Y are independent, we need to check if their joint probability distribution is equal to the product of their marginal probability distributions:

P(X = x, Y = y) = P(X = x) * P(Y = y)

We can start by calculating the marginal probability distributions of X and Y, which are:

P(X = 1) = P(X = 1, Y = -1) + P(X = 1, Y = 0) = 1/6 + 1/3 = 1/2

P(X = 2) = P(X = 2, Y = -1) + P(X = 2, Y = 0) + P(X = 2, Y = 2) = 1/6 + 1/6 + 1/6 = 1/2

P(Y = -1) = P(X = 1, Y = -1) + P(X = 2, Y = -1) = 1/6 + 1/6 = 1/3

P(Y = 0) = P(X = 1, Y = 0) + P(X = 2, Y = 0) = 1/3 + 1/6 = 1/2

P(Y = 2) = P(X = 2, Y = 2) = 1/6

Now we can calculate the joint probability distribution:

P(X = 1, Y = -1) = 1/6

P(X = 2, Y = -1) = 1/6

P(X = 1, Y = 0) = 1/3

P(X = 2, Y = 0) = 1/6

P(X = 2, Y = 2) = 1/6

As we can see,

P(X = 1, Y = -1) = P(X = 1) * P(Y = -1) = (1/2) * (1/3) = 1/6

P(X = 2, Y = -1) = P(X = 2) * P(Y = -1) = (1/2) * (1/3) = 1/6

P(X = 1, Y = 0) = P(X = 1) * P(Y = 0) = (1/2) * (1/2) = 1/4

P(X = 2, Y = 0) = P(X = 2) * P(Y = 0) = (1/2) * (1/2) = 1/4

P(X = 2, Y = 2) = P(X = 2) * P(Y = 2) = (1/2) * (1/6) = 1/12

Therefore, X and Y are independent.

It's important to notice that independence is a stronger condition than the absence of correlation, two random variables can be correlated but independent, for example if you roll two dices, the sum of the numbers is correlated to the numbers on each dice but the numbers on each dice are independent.

In fact X and Y are not independent can you spot the "obvious" mistake?