Positive integral for two functions

In summary, if f(x) is 0 where g(x) is +ve and f(x) > 0 where g(x) is -ve, then the equal sign holds.
  • #1
ChrisVer
Gold Member
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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.
 
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  • #2
ChrisVer said:
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.

Suppose f(x) is 0 where g(x) is +ve. Can you get a counterexample from that?
 
  • #3
If [itex]f(x)=0[/itex] then the equal sign holds...?
 
  • #4
ChrisVer said:
If [itex]f(x)=0[/itex] then the equal sign holds...?

I don't think you understood my idea. f(x) = 0 where g(x) is +ve and f(x) > 0 where g(x) is -ve.

It's not that hard to see.
 
  • #5
I see... those statements are not sufficient then... :/
 

Related to Positive integral for two functions

1. What is a positive integral for two functions?

A positive integral for two functions is a mathematical concept that represents the area between two functions on a graph when the area is above the x-axis and is therefore positive. It is also known as a definite integral.

2. How is a positive integral for two functions calculated?

The calculation of a positive integral for two functions involves taking the definite integral of the function on top and subtracting the definite integral of the function on the bottom. This can be done using various methods such as the fundamental theorem of calculus or integration by parts.

3. What is the significance of a positive integral for two functions?

A positive integral for two functions has several applications in mathematics and science. It can be used to calculate areas under curves, volumes of 3D shapes, and probabilities in statistics. It also has applications in physics, economics, and engineering.

4. Can a positive integral for two functions have a negative value?

No, a positive integral for two functions represents the area above the x-axis and is therefore always positive. If the area between the two functions is below the x-axis, it is considered a negative integral and the value is subtracted from the positive integral to get the overall value.

5. Are there any limitations to calculating a positive integral for two functions?

While the concept of a positive integral for two functions is widely applicable, there are certain limitations to its calculation. For example, if the functions have infinite discontinuities or do not have a closed form, it may not be possible to calculate the integral using standard methods.

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