What is the relation between zero covariance and independence?
Property 2 says that if two variables are independent, then their covariance is zero. This does not always work both ways, that is it does not mean that if the covariance is zero then the variables must be independent.
Why is covariance zero independent?
If X and Y are independent variables, then their covariance is 0: Cov(X, Y ) = E(XY ) − µXµY = E(X)E(Y ) − µXµY = 0 The converse, however, is not always true. Cov(X, Y ) can be 0 for variables that are not inde- pendent. There’s a general formula to deal with their sum when they aren’t independent.
What does it mean to have 0 covariance?
Unlike Variance, which is non-negative, Covariance can be negative or positive (or zero, of course). A positive value of Covariance means that two random variables tend to vary in the same direction, a negative value means that they vary in opposite directions, and a 0 means that they don’t vary together.
Is it true that zero covariance implies independence?
For MVN, zero covariance implies independence. How about other distributions? – Cross Validated According to this page on wikipedia, if X ∼ N ( μ, Σ) with μ ∈ R 2 and Σ ∈ R 2 × 2 then we have Cov ( X 1, X 2) = 0 ⟹ X 1 ⊥ ⊥ X 2. Is this true for any other families of distribution besides the multivariate normal? Know someone who can answer?
What happens if two variables have a covariance of 0?
“If two variables are independent, their covariance is 0. But, having a covariance of 0 does not imply the variables are independent.”. This is nicely explained by Macro here, and in the Wikipedia entry for independence.
Is the covariance of a joint distribution always zero?
Or more generally, take any distribution P ( X) and any P ( Y | X) such that P ( Y = a | X) = P ( Y = − a | X) for all X (i.e., a joint distribution that is symmetric around the x axis), and you will always have zero covariance.
Is the covariance of X and y dependent?
Covariance is zero (and E(XY) = 0, which is the criterion for orthogonality), yet they are dependent. Credit goes to this post. IN PARTICULAR (OP problem): These are Bernoulli rv’s, X and Y with probability of success Pr (X = 1), and Pr (Y = 1).