For $n \in N $, if $X_n \sim Poisson(\frac{1}{n})$ then
PT: 1. $X_n \xrightarrow[n\rightarrow \infty]{P} 0 $
- $nX_n \xrightarrow[n\rightarrow \infty]{P} 0 $
It says $X_n$ converges to 0 in probability.
Attempt:
$ \forall \epsilon >0, $
$Pr(|X_n - 0| > \epsilon ) = 1 - e^{-\lambda} \Sigma_{i=0}^{\lfloor{\epsilon}\rfloor} \frac{\lambda^i} {i!} $
Putting $\lambda = \frac{1}{n}$ in the above equation, 1 is proved by saying as limit of n approaches $\infty$, the summation is a finite sum of quantities approaching 0 therefore it is 0.
Q1. Is there a better way to prove ?
Q2. How can we prove 2 ?