In the last post, I said I’d give an example of two random variables whose expectations are independent but are not themselves independent.
For a first attempt, this turned out not to be an easy task because when coming up with an example I had to control two things: 1) making sure I can get for some and 2) making sure I can get .
I realized that to make like easier I should try to get which means I only have to make sure one of the expectations and I can use to freely adjust to try and make .
Consider a fair six-sided die. Let
Now, after playing around with , I came up with this
Yay! Now, check for non-independence