Covariance of two variables hat check I can't do this, I give up. I am just not able to do it. I don't know what is wrong with me but I can't do it and I need help. Hat check experiment with 3 hats, outcomes 1,2,3|1,3,2|2,1,3|3,2,1 have probability of 1/5, rest are 1/10 I want to find the covariance of $X_1$ = person 1 gets their hat and Y = number of people that get their hat. So I need to find the E(X,Y) which is a cross product multiplication of all the possibilites of the outcomes of both experiments combines. Ok so there are only two outcomes I need to look at for $X_1$ and both have probability of 1/5. There are two outcomes for N, 3 with probabiliy 1/5 and 1 with probability 3/5. This gives me $2/5 * 3/5 + 3/5* 2/5$ This gives me 12/25 So plug in this mindless formula $12/25 - E(X_1)E(Y)$ I know that $E(X_1) = 2/5$ $E(Y) = 6/5$ That is 12/25 so my covariance is zero but this is wrong. Where did I go wrong. Oh lawd jesus someone save me before I go insane.

Try making a table of the events as well a their corresponding outcomes and probabilities. Then define Z = X*Y and find the expected value of Z.


outcome | X | Y | P | Z |

1,2,3 1 3 1/5 3
1,3,2 1 1 1/5 1
etc.


I sincerely hope that my answer made it in time before you went insane!