Limit of probabilities of a large sample

MAXIM LI
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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##
 
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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
$$\lim_{n \to \infty} P(\overline{X}_n = 2)= \lim_{n \to \infty} 0 = 0$$
 
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There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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