Limit of probabilities of a large sample

  • #1
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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  • #2
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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FAQ: Limit of probabilities of a large sample

What is the limit of probabilities in the context of a large sample?

The limit of probabilities in the context of a large sample refers to the behavior of the probability distribution of a sample statistic as the sample size becomes very large. It often involves concepts from the Law of Large Numbers and the Central Limit Theorem, which describe how sample averages and other statistics converge to their expected values or normal distributions as the sample size increases.

How does the Central Limit Theorem relate to the limit of probabilities for large samples?

The Central Limit Theorem (CLT) is a fundamental principle that states that the distribution of the sum (or average) of a large number of independent, identically distributed random variables approaches a normal distribution, regardless of the original distribution of the variables. This theorem is crucial for understanding the limit of probabilities, as it allows us to make inferences about population parameters even when the sample size is large.

What is the Law of Large Numbers and how does it affect probabilities in large samples?

The Law of Large Numbers (LLN) states that as the sample size increases, the sample mean will converge to the population mean. This principle is essential in probability theory because it ensures that the averages of large samples are reliable estimates of the expected value. Therefore, in large samples, the probability of the sample mean being close to the population mean increases.

Why is the normal distribution important in the context of large sample probabilities?

The normal distribution is important because, according to the Central Limit Theorem, the distribution of the sample mean of a large sample will approximate a normal distribution, regardless of the shape of the population distribution. This property simplifies the process of making statistical inferences about the population, as the normal distribution is well understood and has convenient mathematical properties.

How do confidence intervals relate to the limit of probabilities in large samples?

Confidence intervals provide a range of values within which the true population parameter is expected to lie with a certain probability. In the context of large samples, the limit of probabilities ensures that these intervals become more accurate as the sample size increases. The Central Limit Theorem allows us to use the normal distribution to calculate these intervals, making them a practical tool for estimating population parameters from large samples.

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