Is the following statement correct?
If $X$ and $Y$ are independent then $\operatorname{Var}(X)\lt \operatorname{Var}(XY)$ and $\operatorname{Var}(Y)\lt \operatorname{Var}(XY).$
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1Welcome to CV. Since you’re new here, you may want to take our tour, which has information for new users. Since this looks like homework (apologies if it's not), please add the [self-study] tag and read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. Please make these changes as just posting your homework & hoping someone will do it for you is grounds for closing. If this is self-study rather than homework, let us know, and... it's still a good idea to show us what you've tried. – jbowman Jan 10 '20 at 16:25
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For the first inequality, consider what happens to the right hand side $\operatorname{Var}(XY)$ when $Y$ is multiplied by a sequence of values $\sigma_n$ that converge to $0.$ If the inequality is always true, what would that imply about $\operatorname{Var}(X)$? – whuber Jan 10 '20 at 16:42
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Further to comment by jbowman; if it's not homework, please clarify what you're trying to achieve (some background to what prompts the question). If it is homework you can use some basic properties of [expectation](https://en.wikipedia.org/wiki/Expected_value#Linearity) and [variance](https://en.wikipedia.org/wiki/Variance#Basic_properties), where with independence you should be able to expand Var(XY) into a simple expression in terms of means and variances that's symmetric in X and Y. If you then consider centered variables with small variance, the answer should then be obvious. – Glen_b Jan 11 '20 at 05:24
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It is not homework. I am going to derive the CRLB for estimated TOA in the wireless sensor networks. So, I need to know the relation between variances. I studied properties of Variance and expectation but I could not prove the mentioned equation. – Zoha Abdi Jan 13 '20 at 08:15