Charasteristic function of integer valued distribution

[bennyzadir]

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.

 

[CaptainBlack]

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

 

[bennyzadir]

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!