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I am trying to find out expectation of a function of a uniform random variable. I am given a random variable $x$ that is uniformly distributed over the interval $[0, a]$. I want to find out the expectation $E[e^{-i2\pi (m-n)\frac{x}{a}}]$, where $m$, $n$ are integers and $i=\sqrt{-1}$.

I saw derivation at a few places, and it looks like that the expectation evaluates to $\delta_{km}$. It is not clear to me how this derivation is carried out. When I look at the expectation, it appears I am trying to calculate the characteristic function of a uniform random variable. This function is not a delta function. So what am I missing here?

whuber
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ryan80
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  • I am not sure that it is a duplicate. The question was about U[01] and the characteristic function wasn't mentioned. It is only in the long answer of whuber that you see the characteristic function. The OP is using U[0,a]. Maybe that doesn't matter. But could he be talking about a uniform with an unknown parameter a? – Michael R. Chernick Dec 11 '16 at 16:28
  • You can find the derivation [here](https://www.statlect.com/probability-distributions/uniform-distribution). – Antoni Parellada Dec 11 '16 at 16:38

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