I found this characteristic function:
$ E[e^{-\lambda X}] = \sum_{k=0}^{\infty} e^{-\lambda k} P(X=k) $
How can I find P(X=0)?
I found this characteristic function:
$ E[e^{-\lambda X}] = \sum_{k=0}^{\infty} e^{-\lambda k} P(X=k) $
How can I find P(X=0)?
As Henry says, it's the moment generating function of $X$, and its value is taken at $-\lambda$. We can write $E[e^{-\lambda X}]=P(X=0)+\sum_{k=1}^{+\infty}e^{-\lambda k}P(X=k),$ and for $\lambda >0$ $0\leq \sum_{k=1}^{+\infty}e^{-\lambda k}P(X=k)\leq \sum_{k=1}^{+\infty}e^{-\lambda k}=\frac{e^{-\lambda}}{1-e^{-\lambda}}$ so $\lim_{\lambda\to+\infty}E[e^{-\lambda X}]=P(X=0)$ and if you know $E[e^{-\lambda X}]$ and its limit when $\lambda\to +\infty$ is not too hard to compute you can deduce $P(X=0)$.