Xn binomial limit to infinity

Question

Let $X_n$ be random variables with

$X_n \sim \mathcal{B}(n, 2/n)$

Find $\lim_{n \to\infty }(P(X_n=2))$

I don't know where to start, any help is appreciated.

Answer 1

The hint in the other answer, should be enough. Notice that if $X_{n}\sim \mathcal{B}(n, 2/n)$ then $$P(X_n=2)=\binom{n}{2}\frac{2^2}{n^2}\left(1-\frac{2}{n}\right)^{n-2}$$ So you need to calculate $$\lim_{n\to\infty} \frac{n!}{2!(n-2)!}\frac{4}{n^2} \left(1-\frac{2}{n}\right)^{n-2}=\frac{2}{e^2}$$

This limit actually is not that hard, since $$\frac{n!}{2!(n-2)!}\frac{4}{n^2}=\frac{n-1}{n}\frac{n}{n}\cdot 2\to 2,\ \ \mathrm{as}\ n\to\infty $$

and $$\left(1-\frac{2}{n}\right)^{n-2}=\left(1-\frac{2}{n}\right)^{n}\left(1-\frac{2}{n}\right)^{-2}\to e^{-2}\cdot 1\ \ \mathrm{as}\ \ n\to\infty$$

You can check: https://en.wikipedia.org/wiki/Binomial_distribution and for the limit https://www.researchgate.net/profile/Alvaro_Salas3/publication/265777624_The_exponential_function_as_a_limit/links/55f79c4b08aec948c471542e/The-exponential-function-as-a-limit.pdf

Answer 2

HINT

1. Please write down an explicit form for $f(n) = \mathbb{P}[X_n=2]$.
2. Then you can compute explicitly what is $\lim_{n \to \infty} f(n)$.

Answer 3

Hint:

Since $n\cdot \frac 2n = 2$, $X_n$ converges in distribution to the Poisson distribution with $\lambda = 2$.

Hence, $\lim_{n\to\infty}P(X_n=2)= \frac{2^2e^{-\lambda}}{2!} = \frac 2{e^2}$.