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I am feeling stupid today...so:
Is the following statement true?
\int_{-\infty}^{+\infty} f(x) g(x) \ge 0
if f(x) \ge 0 and 0< \int_{-\infty}^{+\infty} g(x) < \infty (integral converges)?
If yes, then how could I show that? (It's not a homework) I am trying to understand how the expectation value E[XY] = <X,Y> for two variables X,Y can indeed be that inner product. Which would need that <X,X>=E[X^2] \ge 0 \Rightarrow \int_{-\infty}^{+\infty} x^2 g(x) dx \ge 0. Here f(x)=x^2 >0 and \int g(x) = 1 (normalized pdf). Obviously it's true since all the negative values of the X will become positive and so their expectation should be positive, but I'm trying to look in it mathematically.
Is the following statement true?
\int_{-\infty}^{+\infty} f(x) g(x) \ge 0
if f(x) \ge 0 and 0< \int_{-\infty}^{+\infty} g(x) < \infty (integral converges)?
If yes, then how could I show that? (It's not a homework) I am trying to understand how the expectation value E[XY] = <X,Y> for two variables X,Y can indeed be that inner product. Which would need that <X,X>=E[X^2] \ge 0 \Rightarrow \int_{-\infty}^{+\infty} x^2 g(x) dx \ge 0. Here f(x)=x^2 >0 and \int g(x) = 1 (normalized pdf). Obviously it's true since all the negative values of the X will become positive and so their expectation should be positive, but I'm trying to look in it mathematically.
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