- #1

ChrisVer

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I am feeling stupid today...so:

Is the following statement true?

[itex] \int_{-\infty}^{+\infty} f(x) g(x) \ge 0 [/itex]

if [itex] f(x) \ge 0 [/itex] and [itex]0< \int_{-\infty}^{+\infty} g(x) < \infty [/itex] (integral converges)?

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

Is the following statement true?

[itex] \int_{-\infty}^{+\infty} f(x) g(x) \ge 0 [/itex]

if [itex] f(x) \ge 0 [/itex] and [itex]0< \int_{-\infty}^{+\infty} g(x) < \infty [/itex] (integral converges)?

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

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