Limit of probabilities of a large sample

[MAXIM LI]

Homework Statement

## Let {X_n}_{n≥1}## be a sequence of iid random variables having a common density function
## f(x) = \begin{cases} xe^{-x} &\text{ for } x \ge 0 \\ 0 &\text{ otherwise }\end{cases}##

Let ##\bar{X}_n = \frac{1}{n}\sum_{i=1}^{n} X_i## where ##n=1,2,\ldots##. Then find ##\lim_{{n\to\infty}} P(\bar{X}_n=2)##

Relevant Equations

##\lim_{{n\to\infty}} P(\bar{X}_n=2)##

My first thought as well but I think the problem is deeper than that. I think that as the n tends towards infinity the probability of the the sample mean converging to the population mean is 1. Looking at proving this.
By the Central Limit Theorem the sample mean distribution can be approximated by a Normal distribution with $$\mu = 2,~\sigma = \sqrt{\dfrac{2}{n}}$$

As ##n\to \infty## this becomes a delta function centered at ##2##

 

[Aryan1043]

You're overcomplicating this. ##\overline{X}_n## is a continuous random variable so ##P(\overline{X}_n = a) = 0## for all ##a \in \mathbb{R}##. In particular
$$\lim_{n \to \infty} P(\overline{X}_n = 2)= \lim_{n \to \infty} 0 = 0$$