MHB Charasteristic function of integer valued distribution

AI Thread Summary
The discussion focuses on proving that the probability mass function of an integer-valued distribution can be derived from its characteristic function using the formula p(k) = (1/2π) ∫ from -π to π e^(-ikt)φ(t) dt for all k in Z. Key to this proof is the integral property that states ∫ from -π to π e^(-ikt)e^(ilt) dt equals 2π when k equals l, and 0 otherwise. The approach involves changing the order of integration and summation in the integral of the characteristic function. Participants express gratitude for assistance in understanding this mathematical concept. The conversation emphasizes the relationship between characteristic functions and probability mass functions in integer-valued distributions.
bennyzadir
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How to prove that if $\varphi$ is the characteristic function of an integer valued distribution, then the probability mass function can be computed as
$ p(k) = \frac{1}{2\pi} \cdot \int^{\pi}_{-\pi} e^{-ikt}\varphi(t) dt \;,\forall k \in \mathbb{Z} $

I would be really grateful if you could help me.
 
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zadir said:
How to prove that if $\varphi$ is the characteristic function of an integer valued distribution, then the probability mass function can be computed as
$ p(k) = \frac{1}{2\pi} \cdot \int^{\pi}_{-\pi} e^{-ikt}\varphi(t) dt \;,\forall k \in \mathbb{Z} $

I would be really grateful if you could help me.

You need to know that:

\( \displaystyle \int_{-\pi}^{\pi} e^{-ikt}e^{ilt}\; dt=2 \pi\) if \(k=l\) and \(0\) otherwise.

Then you just change the order of integration and summation in \( \int_{-pi}^{\pi} e^{-kt}\phi(t)\; dt\) to get the required result.

CB
 
CaptainBlack said:
You need to know that:

\( \displaystyle \int_{-\pi}^{\pi} e^{-ikt}e^{ilt}\; dt=2 \pi\) if \(k=l\) and \(0\) otherwise.

Then you just change the order of integration and summation in \( \int_{-pi}^{\pi} e^{-kt}\phi(t)\; dt\) to get the required result.

CB

Thank you so much!
 
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