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### Is the cosine angle between two R.V. an (approximation) not equality to the correlation coefficient?

Hint: $X \cdot Y$ is a random variable. $\text{Cov}(X,Y)$ is an expected value. ---- addendum ----* There is some confusion of terminolgy in your post. If $\bf X , \bf Y$ are two random vectors (in ...
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### Covariance, and the Taylor expansion for the expected value of a linear function of random variables

Expectation is linear, and therefore $$\mathbb{E}[\theta] = \mathbb{E}[U(X)]-\mathbb{E}[U(Y)]$$ regardless of whether $U,Y$ are independent or not. They could be arbitrarily correlated, this would ...
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