# Charasteristic function of integer valued distribution

##### New member
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

##### Well-known member
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

• $$\displaystyle \int_{-\pi}^{\pi} e^{-ikt}e^{ilt}\; dt=2 \pi$$ 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.