Covariance of two random variables

I wanted to ask can someone explain to me how to prove this implication:
If Y is a random variable that has finite second moments and distribution symmetric about 0, then the covariance of Y and the absolute value of Y is zero.
Any help you can provide is greatly appreciated, thank you in advance!

p.lucia.p said:

I wanted to ask can someone explain to me how to prove this implication:
If Y is a random variable that has finite second moments and distribution symmetric about 0, then the covariance of Y and the absolute value of Y is zero.
Any help you can provide is greatly appreciated, thank you in advance!

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The question doesn't makes sense to me. Covariance is something for comparing two random distributions and you have mentioned one random variable. As far as an absolute value of Y being zero, that would contradict the fact that it is a random distribution since the sum (definite integration) must be 1 and the sum (definite integral) of 0 is zero.