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I am trying to understand these two definitions, but I'm just not clear where they come from and how they relate to the old formulas that I was used to a week ago.

In one book variance is defined as: $\displaystyle Var(X) = \int (x-\mu)^2 f(x) dx $ and in the other variance is $E[(X-\mu)^2]$. How are these two the same? Can someone also explain this formula: $\displaystyle \mu = \int xf(x) dx$?

Person
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  • Where did you find the definition of the variance as $E[(x-\mu)^2]$? That is, can you provide a specific reference (book author(s), title, publisher, and possibly page number)? Be very very careful in making sure that you do NOT mix $X$ and $x$ when you copy the formula from that book; they have different meanings. – Dilip Sarwate Aug 13 '13 at 21:24
  • I corrected the typo. Yes I do know that $X$ represents a random variable. – Person Aug 13 '13 at 21:30
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    Read about the law of the unconscious statistician to understand why $$E[g(X)] = \int_{-\infty}^\infty g(x)f_X(x),\mathrm dx$$ for a continuous random variable (other kinds too if care is taken in the definition) in general. Then, set $g(X) = (X-\mu)^2]$ – Dilip Sarwate Aug 13 '13 at 21:37

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