Why are $E[X]_, \;E[Y]= 0$ if $X_,\; Y$ are measured from their means?

Question

I was studying some examples of linear correlations where I got the reasoning:

Since $X_,\;Y$ are measured from their means, $$E[X]= 0=E[Y]$$ ....

Can anyone tell me why it is so?

Answer 1

To see what "measured from the means" means, consider the following example. Let $X$ be the random variable that measures the temperature in a place with mean temperature $E[X]=20^o$C. Let the temperature be $X=21$ at some given day. You can either write $X=21^o$ or $X=21-20=1$. In the second case you "measure $X$ from its mean". Similarly $X=19^o$ can be written as $X=-2$ etc. So, you use a new variable $$X'=X-μ$$ where $μ=Ε[Χ]$. Obviously $$E[X']=E[X-μ]=E[X]-E[X]=0$$ Τo avoid unnecessary notation, the author does not denote this new variable with $X'$ and just says that $X$ is "measured from its mean". Similarly for $Y$. You may also encounter the term "centered random variables". It is the same.

Answer 2

I think means than the axis is centerded on their means. Think about if we know the mean of a r.v. we can center the r.v. by substracting the mean.