- #1

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## Homework Statement

Given that the expected value of a function is equal to the expected value of the convolution of that function

*with a transfer function*

**f**

**g**[tex]

E(f) = E(\widetilde{f})

[/tex]

prove that the following holds

[tex]

\int^{+\infty}_{-\infty} g(t) \, dt = 1

[/tex]

Here

*is the function and*

**f***is the transfer function, and [tex]\widetilde{f}=f*g[/tex] is the convolution*

**g**## Homework Equations

[tex]

\widetilde{f} = \int^{+\infty}_{-\infty} f(x-t)g(t) \, dt

[/tex]

[tex]

E(f) = \int^{+\infty}_{-\infty} xf(x) \, dx

[/tex]

## The Attempt at a Solution

I have no idea how to approach this, my guess is that by replacing

*on both sides would lead to the proof:*

**E(f)**[tex]

\int^{+\infty}_{-\infty} xf(x) \, dx = \int^{+\infty}_{-\infty}\int^{+\infty}_{-\infty} xf(x-t)g(t) \, dt\, dx

[/tex]