# How to derive to this point

## Main Question or Discussion Point

hello, can someone look at my problem and tell me how to get to the arrow?
thank you so much..
i am studying for my midterm and i have no clue where it come from.
thanks

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uart
That first line you point to is basically just the definition of how you find the "expected value" of a function.

That is, if a random variable X has a pdf (probablity density function) of f(x) then the expected value of x can be written as,

$$E(x) = \int_{-\infty}^{+\infty} x f(x) dx$$

And more generally the expected value of a function of x can be written as,

$$E(\, \phi(x)\, ) = \int_{-\infty}^{+\infty} \phi(x) f(x) dx$$

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uart
BTW, there's an error in that derivation. The x^2 term that appears in the first square bracketed term in line 4 (2nd line under the completing the square heading) should be just "x" (not squared). This error is carried all the way through the derivation BTW.

yeah.i understand that part..
but how does exp(-o^2w^2/2) = exp(-x^2/2o^2)?
the arrow....

uart
yeah.i understand that part..
but how does exp(-o^2w^2/2) = exp(-x^2/2o^2)?
the arrow....
What? It never says that anywhere. The question says to prove that,

$$\Phi(\omega) = exp(-\sigma^2 \omega^2 /2)$$

You must be misreading it because nowhere does it claim the thing you state.

HallsofIvy
$$e^{\frac{-\sigma^2\omega^2}{2}}[/itex] and them immediately uses the fact that the Gaussian distribution (with mean 0) itself can be written as [tex]e^{\frac{-x^2}{2\sigma^2}$$