Solutions - 5130-01

(1) State CDF of a Poisson distribution:

We know that $P_X(k)=e^{-\alpha }\frac{{\alpha }^k}{k!}$

We know that $F_X(k)=P[X\le k]$.

$$F_X(k)=\sum _{k=0}^{\infty }{e}^{-\alpha }\frac{{\alpha }^k}{k!}$$

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(2) Compute limit as $k\to \infty$:

Note that $e^x={\sum }_{n=0}^{\infty }\frac{x^n}{n!}$

$$\underset{k\to \infty }{\lim }{F}_X(x)=e^{-\alpha }\underset{k\to \infty }{\lim }\sum _{k=0}^{\infty }\frac{{\alpha }^k}{k!}=e^{-\alpha }{e}^{\alpha }=1$$