query: Are there any counterexamples which show that $E(XY) = E(X)E(Y)$ cannot imply $X,Y$ is not necessarily independent?
positive_passages.docid: 153393
positive_passages.text: Is there a *simple* example showing that uncorrelated random variables need not be independent?
negative_passages.docid: 107888
negative_passages.text: Why zero correlation does not necessarily imply independence
negative_passages.docid: 136707
negative_passages.text: Show that if $X$ and $Y$ are independent with the same exponential distribution, then $Z= |X - Y|$ has the same exponential distribution
negative_passages.docid: 19973
negative_passages.text: If $X$ and $Y$ are independent $N(0,\sigma^2)$, then $X^2+Y^2$ and $X/Y$ are independent?
negative_passages.docid: 54461
negative_passages.text: Showing independence of random variables
negative_passages.docid: 135641
negative_passages.text: Algebraic independence of exponentials
negative_passages.docid: 48270
negative_passages.text: How to prove that the random variables $X$ and $X^2$ are not independent?
negative_passages.docid: 121558
negative_passages.text: Showing that $n$ exponential functions are linearly independent.
negative_passages.docid: 433
negative_passages.text: Problem about jointly continuous and linearity of expectation.
negative_passages.docid: 125209
negative_passages.text: Prove $e^x - e^y \leq e |x-y|$ for $x$ belonging to $[0,1]$
negative_passages.docid: 62007
negative_passages.text: Durrett Example 1.9 - Pairwise independence does not imply mutual independence?
negative_passages.docid: 118867
negative_passages.text: Independence of Gaussian variables
negative_passages.docid: 16673
negative_passages.text: If $X$ and $Y$ are independent then $f(X)$ and $g(Y)$ are also independent.
negative_passages.docid: 143732
negative_passages.text: Prove random variable $Y_0$ and $\sigma$-algebra $\mathscr{R}$ are independent.
negative_passages.docid: 88431
negative_passages.text: Jointly Gaussian uncorrelated random variables are independent
negative_passages.docid: 53270
negative_passages.text: Intuition/Real-life Examples: Pairwise Independence vs (Mutual) Independence
negative_passages.docid: 34527
negative_passages.text: How to show a sequence of independent random variables do not almost surely converge by definition?