Correlation coefficient of a jumping particle

[Addez123]

Homework Statement

Assume the a particle to be in origo. Every step it jumps either -1 or 1 in either x or y plane.
So for each step it can move (1,0), (-1,0), (0,1) or (0,-1).
Let the particle start in origo and X, Y define the position after the first step.

Calculate ρ(X, Y)

Relevant Equations

ρ(X, Y) = C(X, Y) / (D(X) * D(Y))
C = Covariance
D = Standard deviation

C(X, Y) = E(XY) - E(X)E(Y)

What I need help with is how I would start..
I can say p(X, Y) = (1,0) = 1/4, and same for the other 3 coordinates. P = 0 for all other coordinates.
This doesn't give me anything to work with though.

C(X, Y) = E(XY) - E(X)E(Y)
What is XY? I don't even know what X is.

 

[haruspex]

What is the probability distribution of X?

 

[Addez123]

Maybe 0 if x < -1, jumps to 1/2 if x = -1 then jumps to 1 at 1?
Idk how to express it in numbers though.

 

[Delta2]

Maybe 0 if x < -1, jumps to 1/2 if x = -1 then jumps to 1 at 1?
Idk how to express it in numbers though.

No I don't think its like that, I expect it to be symmetric around x=0, if by probability distribution we mean the probability $P(X=x)$ where x an integer. The particle can move to +1 or -1 of its current position with respect to x-axis. So it has a non zero probability to be found at any $(x,y)$ where (x,y) integers.

@haruspex any hints how to calculate the probability P(X=x) cause I am stuck also and my probabilities are rusty. The problem I am seeing is that there are infinite ways in which the particle can arrive at (x,y). Am I overcomplicating the issue (I probably am but I don't know where my thinking is wrong).

EDIT: I should have read the problems statement more carefully. X,Y are defined in a specific different way.

 

[haruspex]

Maybe 0 if x < -1, jumps to 1/2 if x = -1 then jumps to 1 at 1?
Idk how to express it in numbers though.

(X, Y) is defined as the position after one step. So it is equally likely at each of four positions. In how many of those is X=-1, etc.?

 

[Addez123]

I was able to calculate
$$E(x) = \sum_k g(k)*p_x(k) = -1 * 1/4 + 1 * 1/4 = 0$$
Then I got variance by
$$V(x) = E(x^2) - E(x)^2 = 1*1/4 + 1*1/4 - 0 = 1/2$$
$$D(x) = \sqrt{V(x)} = 1/\sqrt{2}$$
Same for y since they are identical.
$$ρ(X, Y) = C(X, Y) / (D(X) * D(Y)) $$
$$C(X, Y) = \sum_j \sum_k (j - ux)(k - uy)p(j, k) = 1/4 * (1*0 + -1*0 + 0*1 + 0*-1) = 0$$
Now the second question is: Are they independent?

X and Y are independent if
$$E(XY) = E(X)E(Y)$$
We can get E(XY) from this equation:
$$C(X,Y) = E(XY) - E(X)E(Y)$$
$$E(XY) = -E(X)E(Y) - C(X,Y) = -0*0 -0 = 0$$
Therefore x and y are independent. My book says they are NOT independent though.

Where did I do wrong?

 

[haruspex]

X and Y are independent if $E(XY) = E(X)E(Y)$

If X and Y are independent then that is true, but it is not a sufficient condition for independence. Check your notes.