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Questions tagged [central-limit-theorem]

This tag should be used for each question where the term "central limit theorem" and with the tag (tag:probability-limit-theorems).

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1answer
63 views

How to apply the central Limit theorem?

I am interested on upper bounding the following probability as $n$ goes to infinity. \begin{equation} \mathbb{P} \left\lbrace \Big|( \xi_{n}- \mathbb{E} \xi_{n})\Big|> \ell \right\rbrace \end{...
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1answer
1k views

When to use the continuity correction for normal approximations of binomial probabilities.

so I'm confused as to when you actually use continuity correction. If a problem deals with a binomial distribution and we are asked to find probabilities using normal approximation (provided np>5 and ...
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1answer
32 views

LLN and CLT on $Y_n(t) := \boldsymbol{1}_{X_n \le t}$

Let $X_n$ be a sequence of i.i.d. random variables where $X_1$ has distribution function $F$. Fix $t \in \Bbb R$. Define $Y_n(t) := \boldsymbol{1}_{X_n \le t}$. Certainly, the $Y_n(t)$ are again i.i.d....
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0answers
62 views

Lindeberg's Condition is not necessary

Let $X_n^k$ be Triangular Array ($k\leq n$) of independent mean $0$ random variables. Suppose $\sum_k^n \text{Var}(X_n^k) = 1$. Lindeberg's Condition \begin{equation} \lim_{n\to\infty} \sum_k^n ...
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1answer
85 views

A failure of convergence of conditional distributions

Consider $n$ iid samples $(X_i,Y_i)$ generated such that $X_i$ is a truncated normal of mean $\mu_X$ truncated to the left of the origin, and $Y_i$ is a truncated normal of mean $\mu_Y$ truncated to ...
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1answer
34 views

CLT with random (Binomial) number of summands

Say we have iid real-valued $X_1,X_2,\ldots$ with expectation $\theta\in\mathbb{R}$ and variance $\sigma^2>0$. Then clearly $$\sqrt{n}\left(\frac{1}{n}\sum_{i=1}^n X_i-\theta\right)\xrightarrow[n\...
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2answers
177 views

Question regarding central limit theorem proof using Lyapunov's condition

Let $X_n\sim \operatorname{independent} \operatorname{Uniform} (0,n^2)$. I need to find $a_n$ and $b_n$ such that $$\frac{\sum_{k=1}^n (X_k -a_n)}{b_n}$$ converges in distribution to a non-degenerate ...
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1answer
26 views

Show as $\ n\rightarrow\infty\ $, $\ \sqrt{n}(Y_n-p)\rightarrow N(0,p(1-p))$

Let $X_i$, $i=1,2...,$ be independent Bernoulli($p$) random variables and let $Y_n=\frac{1}{n}\sum\limits_{i-1}^{n} X_i$. Show that as $n\rightarrow\infty$, $\sqrt{n}(Y_n-p)\rightarrow N(0,p(1-p))$ ...
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1answer
45 views

Calculating integral using gamma distribution

I've been studying form my Probability theory exam and I found this problem: Calculate using Central limit theorem $$\lim_{n\rightarrow\infty}\int_{0}^{n}\frac{1}{(n-1)!}x^{n-1}e^{-x}dx.$$ Using $$\...
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1answer
74 views

Question about central limit theorem on two exercises.

I was presented two formulas, if random variables ${X_1,...,X_n}$ form a random sample of a distribution of mean $\mu$ and standard deviation $\sigma$ and $n \to \infty$: \begin{align} &P\left(\...
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1answer
60 views

Central Limit Theorem on Independent random variables

The Prompt asks me to use Central limit theorem to compute $$P(\sum_{n=1}^{300} X_n > -21)$$ Where F(t) is the distribution function for random variables $X_n,$ for $n = 1, 2, \dots, 300$. $$F(t) = ...
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2answers
113 views

Central limit theorem hotel reservation problem

The prompt There are 100 rooms in a hotel. Since the owner knows that 10% of the early reservations are canceled before the arrival, he ordered to accept reservations for more than 100 rooms. What is ...
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0answers
29 views

Cumulative distribution function of the lifetime of a device

Let $Z$ denote the lifetime of an electrical device. $Z$ is exponentially distributed with parameter $\lambda$. The time it is used daily is denoted by $X_n$ where $X_n$ are iid random variables which ...
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0answers
174 views

Remainder of Taylor series and CLT proof

I'm trying to understand part of the proof of the Central Limit Theorem in Casella Berger 2E. I realize there are many functionally equivalent ways to prove the CLT, but there's a specific step the ...
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0answers
60 views

Compute limits using central limit theorem

I want to find the limit of expressions such as: $\lim_{n \to \infty} P(\frac{|S_{n}|}{n} \le \epsilon) $. I do not know how to proceed. Using the central limit theorem, I can rewrite the expression: ...
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1answer
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The distribution of the sum of i.i.d random variables, each of which equally likely takes value from 1 to 5

Recently, I got stuck with the following task The player twists 50 times a roulette wheel (a circle with 5 sectors of equal size, on these sectors there are numbers from 1 to 5) and each time ...
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1answer
64 views

Confidence Interval and Central Limit Theorem

Here's the problem: "During a test, we asked $120$ people who Jean-Jaques Rousseau was, and $12$ of them answered that he was a driver (which is false). Estimate the proportion of the population ...
2
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1answer
64 views

Show $\sup_x \sqrt{n}|F_n(x) -F(x)|=O_p(1)$ where $F_n$ is the empirical distribution function of a iid RVs with a density function.

Let $X_n$ iid RVs with the distribution function $F$ which has a density. Let $F_n(x)= \frac{1}{n} \sum_{i=1}^nI(X_i \leq x)$ be the empirical distribution function. Show $$sup_x \sqrt{n}|F_n(x) -F(...
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0answers
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falling blocks and the central limit theorem

I'm reading these notes and would like some help deriving the result at the bottom of page 2. Suppose that you are building a tower out of unit blocks. Blocks are falling from the sky, [increasing ...
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1answer
45 views

Is it possible to answer this problem with standard Central Limit Theorem or should we use Lindeberg-Feller CLT?

I have the following problem on my Statistics I problem set: Suppose that $X_t = \mu + U_t$, where $U_t = V_t + \rho V_{t-1}$ and $V_t$ are iid standard normal variables. Apply a CLT to ...
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1answer
90 views

Problem with central limit theorem

Let ($\xi_{k}$)$_{k \in \mathbb{N}}$ be a sequence of i.i.d random variables with expectation $\mu$ and variance $\sigma^2$ $\in (0, \infty)$. We define $X_k$ = $\xi_k$ - $3\xi_{k+1}$ + $\xi_{k+2}$, $...
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1answer
63 views

Proving CLT on a random variable by working with its square

Excuse me if this question is too basic or vague. I have a continuous real random variable say $X$ that is a non-trivial non-explicit function of several independent random variables. My goal is to ...
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1answer
53 views

Estimation of an average, and speed of convergence

I asked myself those two questions the other day, and I have a very limited backgroud in stats, so help would be appreciated! Sometimes in the middle of my grading, I look at the average of the ...
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0answers
27 views

Standardising maximum of Uniform distribution

Let $M_n = \max(U_1,\ldots,U_n)$ , the maximum of a sample of size n from $U(0,1)$ distribution. We want to see what happens with the distribution of $M_n$ (properly standardised or normalised) as $n ...
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0answers
59 views

Show that $\sqrt{\sum_{i}^n X_i} - \sqrt{n} \to Y \sim \mathcal{N}(\mu, \sigma^2)$ in distribution.

Let $(X_n)_n$ be a sequence of i.i.d. random variables with $\mathbb{E}X_1 = 1, \operatorname{Var}(X_1) = 3$. Show that $\sqrt{\sum_{i}^n X_i} - \sqrt{n} \to Y \sim \mathcal{N}(\mu, \sigma^2)$...
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1answer
25 views

Does the probability that at least one statistic will exhibit a “statistically significant” fluctuation converge to 1 as $N\rightarrow\infty$?

In the study and usage of statistics the idea that particular statistics will converge "almost certainly" to some value as the sample size $N$ diverges plays a key role (e.g. the central limit theorem,...
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31 views

How does the first moment affect convergence in distribution?

I have the following problem on my Statistics Problem Set: I was able to use the multivariate Delta Method and solve items a) and b). However, I am struggling with items c) and d). Let me present my ...
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0answers
38 views

Central Limit Theorem Fourier Isometry Confusion

Okay, so I'm trying to understand the proof of the classical CLT. I'm not really a probability guy, so I'll apologize in advance for the analysis-heavy language used. Statement: If $X_i\in L^1\cap L^...
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1answer
41 views

Generalization of the Difference Quotients

Based on the geometry of the three difference quotients in calculus, it seems there should be a general form $$P(f(x + h) - f(x)) + (1 - P)(f(x) - f(x - h))\over h$$ where $0 <= P <= 1$, at ...
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0answers
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Usage of Central Limit Theorem

A machine fills tins with paint. The mass of a tin is normally distributed with a mean of 0.3 kg and a standard deviation of 0.02 kg. The mass of the paint that goes into the tin is normally ...
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1answer
60 views

How to apply the Central Limit Theorem to a sample Poisson distribution

I have a random variable, $X$, that follows a population distribution of the Poisson type, $Po(\lambda)$, with an unknown $\lambda$ parameter. In an experiment, $k=100$ events were measured in a given ...
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2answers
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Central limit theorem Question about working light bulbs

A person has $100$ light bulbs whose lifetimes are independent exponentials with mean $5$ hours. If the bulbs are used one at a time, with a failed bulb being replaced immediately by a new one, ...
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1answer
71 views

Variance of the sum of random variables, each having a mean of 1?

Let $X_1,...,X_{20}$ be independent Poisson random variables with mean 1. Use the central limit theorem to approximate $$P\{\sum_{i=1}^{20} X_i \geq\ 15\}$$. To solve this question, I need to find ...
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26 views

finding probabillity for average value ( continous density function)

Suppose i have a continuos density function for a random variable x as such( function describes runtime for a program in seconds if it means anything): $$ f(x) = \begin{cases} \left(\frac{3}{64}\...
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1answer
38 views

Probabalistic way to calculate limit of a sum [duplicate]

I am following a course on stochastic processes. One question is to Calculate $$lim_{n\to \infty} \sum^n_{i=0} e^{-n} \frac{n^i}{i!}$$ I see that it looks like the expected value of $e^{-n} \frac{n^{...
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1answer
268 views

Intuition for $N(\mu, \sigma^2)$ in terms of its infinite expansion

To gain deeper insight to the Poisson and exponential random variables, I found that I could derive the random variables as follows: I consider an experiment which consists of a continuum of trials ...
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1answer
111 views

central limit theorem stronger condition

In the central Limit Theorem it must be the case that the variance of $X_{i}$, where $\{X_{i}\}_{i \in \mathbb{N}}$ are i.i.d. random variables, must be finite. But can we find independent random ...
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Central limit theorem with multiple samples

Hi I was wondering about the following problem: If there are $1000$ students in a school and the combined scores of these students over the last $5$ years are $5823, 6107, 5672, 6233$ and $5598$ ...
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0answers
62 views

Taylor series of $\log(1+ix)$ and $e^{ix}$

In this proof about a central limit theorem (see Theorem 2.6.2), the author writes the Taylor series for $\log(1+ix)$ and $e^{ix}$ and use it for the calculation for the complex random variable $e^{...
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1answer
60 views

Central Limit Theorem, Does $X_n (\omega)$ Converge Almost Surely?

I'm studying for a final exam I have tomorrow and have a question about the following problem. Let $\Omega = [0,1]$ and let $P([a, b]) = b-a$ for all $0\le a\le b\le 1$. Let $X(\omega) = 0 $ for ...
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1answer
61 views

Central Limit Theorem for Non-degenerate U-Statistics

Let $X_1,X_2,...$ be i.i.d. random variables and $f\colon\mathbb R^{r} \rightarrow \mathbb R$ be a symmetric function of $r$-variables. For each $n \ge r$, the associated U-statistic is defined as, $$...
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1answer
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Problem on Central limit theorem and Law of large numbers

Let $X_n$ be independent random variables which with probability $\frac 1 {2^{n+1}}$ is equal to $2^n$, with probability $\frac 1 {2^{n+1}}$ is equal to $-2^n$ and with probability $1 - \frac 1 {2^{n}}...
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2answers
37 views

Interpreting estimated percentage as a distribution

Imagine I am trying to determine the percentage $p$ of people in the US who voted for the democrats (or republicans, if you prefer). I can determine this by the following process: Randomly ...
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0answers
83 views

Multidimensional CLT for Exchangeable Random Variables

From Wikipedia Let $\mathbf{X}_1, \mathbf{X}_2, ..., \mathbf{X}_n$ be independent $d$ dimensional random vectors having $0$ mean. Let $\mathbf{S}=\sum_{i=1}^n \mathbf{X}_i$ and $\mathbf{\Sigma}$ be ...
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1answer
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Help finding central limit theorem approximations - Normal Distribution Equation

I was given f(x)=|x| as a probability distribution. I've summed the results of a Monte Carlo with N terms and plotted a thousand of these sums in a normalized histogram. Now I need to compare this ...
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1answer
68 views

Confidence intervals vs law of the iterated logarithm

Let $X_i$ be iid random variables with mean $\mu$ and variance $1$. Let $S_n=\frac 1n \sum_{k=1}^n X_k$ denote the sample mean. By CLT it is well known that $(S_n - \frac{1.96}{\sqrt n}, S_n + \frac{1....
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1answer
34 views

convergence and concentration inequalities

I am trying to solve the following problem: Let $\lambda>0$ and $(X_n, n\geq 1)$ be a sequence of i.i.d. r.v. with $X_i\sim \mathcal{E}(\lambda)$. Define as $Y_n= e^{X_n}, S_n = \sum_{j=1}^{n} ...
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2answers
885 views

Central limit theorem for dependent random variables with covariance condition

Consider a sequence of identically distributed real-valued random variables $(X_i)_{i\in\mathbb{N}}$, with $\mathbb{E}\left[X_i\right]=0$ and $\mathbb{E}\left[X_i^2\right]=1$. Suppose that there ...
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1answer
124 views

Calculating Probability and Standard Deviation with Central Limit Theorem

This is another statistical question that I cannot fully understand: Suppose that $100$ fair dice are tossed. Estimate the probability that the sum of the faces showing exceeds 370. Now that the ...
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1answer
18 views

Calculation of Standard Deviation of Sample Mean

This is another statistics problem that I have, which I cannot make sense of: An IQ-test is normal to $ \mu = 100$ and $\sigma = 10$. What is the standard deviation of the sample mean of a sample ...