I’m kind of having trouble with coming up with an example/thought experiment that could explain this. I tried to look around for a simpler explanation but wasn’t really able to find any. I do know that two independent variables have a covariance of 0 but why is it that the other way around does not work?
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This isn’t very realistic but suppose that you live in a city where some people have one dog and others have two dogs. All the people with one dog also own one cat. Meanwhile those with two dogs are split half and half between having no cats and having two cats.
If you knew how many dogs somebody has, then on average, they’d always have one cat. This implies the covariance is 0 since the value of one variable doesn’t effect the average of the other variable. But they are not independent, since knowing the number of dogs someone has does effect the probabilities of the number of cats they have.
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