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). The central limit theorem states that the sampling distribution of the mean of any independent, random variable will be normal or nearly normal, if the sample size is large enough.

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21 views

What is the probability i see at least 900 car in 61 days [closed]

(Xi: I=1,... 365) is a succession of random variables where xi indicates the number of cars I saw on the day i of the year, supposing that the random variables are independent and are identically ...
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19 views

Non-standardised central limit theorem

Let $(X_n)_{n}$ be a sequence of i.i.d. real random variables with mean $\mu \in \mathbb{R}$ and variance $\sigma^2 > 0$. Let $S_n = X_1 + ... + X_n$. The usual central limit theorem ensures that $\...
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1answer
36 views

Show that $P\left ( \left | \sum_{i=1}^{n}X_i \right |\leq 2\sqrt{n} \right )\to 1-2\Phi(-2)$ - solution explanation

Show that $$P\left ( \left | \sum_{i=1}^{n}X_i \right |\leq 2\sqrt{n} \right )\to 1-2\Phi(-2)$$ I have been given that $EX_i=0$ and $Var(X_i)=1$. Now use CLT and thus $$P\left ( \left| \sum_{i=1}^{n}...
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1answer
154 views

Stuck on proving a variation of the Central Limit Theorem.

I'm trying to prove the following version of the central limit theorem. Let $$L^n = (L_1^n,...L_n^n) $$ such that the $L_i$ are i.i.d., there exists a sequence of constants such that $|L_i^n|\leq K^n$ ...
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46 views

Central limit theorem : manufacture of chocolates

In the manufacture of chocolates of a specific variety with a nominal weight of 20.4 grams, there may be fluctuations in the actual weight of a praline. We describe the weight of a praline of this ...
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1answer
49 views

Central limit theorem : game of chance

We consider the following game of chance with independent rounds: In each round we can win ten euros with probability $0.1$, we can lose one euro with probability $0.7$ and two euros with probability $...
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12 views

local behavior of berry-esseen

I am exploring a problem that makes use of a multidimensional Berry-Esseen result for a sum of i.i.d distribution of some positive semi-definite covariance. I have been working with Corollary 8 from ...
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42 views

An unusual setup where sampling without replacement from an infinite sequence might lead to a central limit theorem

Let $a_1, ..., a_N$ denote the alternating sequence $-1, 1, -1, 1, ..., -1, 1$ (assume $N$ is even for simplicity). Let $x_1, ..., x_N$ denote a sequence obtained by random sampling without ...
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Central Limit Theorem - Asymptotic approximation to sums of Poisson random variables and Binomial random variables

Question: Let $X_1, X_2, ...$ be a sequence of iid $\textrm{Poi}(5)$ variables. Let $Y_1, Y_2, ...$ be a sequence of iid $\textrm{Bin}(10,\dfrac{1}{2})$ variables. Define the sums $S_n = \sum_1^n X_i$ ...
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57 views

What is the correct standard deviation when splitting a sample?

I roll a four-faced die 1000 times, but I have 100 dies, so I seperate into 10 rolls of 100 each and tally the result. I want to calculate the standard deviation of the ...
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45 views

Why Central limit theorem is not giving the correct Level of significance?

Consider the following problem: A company manufacturing RAM chips claims the defective rate of the population is $5 \%$. Let $p$ denote the true defective probability. We want to test if: $H_{0}: p=0....
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Multivariate Central Limit Theorem for non-iid case

I already know about the classical Central Limit Theorem (CLT): Let $X_1, \dots , X_n \in \mathbb{R}^d$ be the iid random variables drawn from a distribution with mean $\mu$ and covariance matrix $\...
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Ornstein-Uhlenbeck Bridge as a Random Walk Limit (The Urn Game)

An urn contains $N$ red balls and $N$ black balls. Consider the game in which you sequentially draw balls from the urn: a) with replacement; b) without replacement, until the $2N$ balls are all drawn; ...
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1answer
44 views

Does the Zeta Distribution converge to normal as N gets large

I am curious if the zeta distribution converges to normal if it is summed over many times. I am particularly curious if this is true for $\zeta$(4). I know that, if it does converge to normal, it ...
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46 views

Compute the next limit using CLT

Let $\{X_n\}_n$ be a sequence of random independent variables with density function $f_{X_n}=4x^2e^{-2x} \chi_{x>0}$, $\forall n \in \mathbb{N}$ if $S_n=\sum_{i=1}^n X_i$ find $$\lim_{n \to \infty} ...
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67 views

Probabilities , Central limit theorem

So I am working on a problem where I have to find the approximate law of $\frac{1}{\bar{X}}$ where $X_{j}$ follow the Normal distribution law. Everything is fine up until the point I have to check the ...
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43 views

Central limit problem and converges

Let $\left(X_{j}\right)_{j \geq 1}$ be i.i.d. with $E\left\{X_{j}\right\}=1$ and $\sigma_{X_{j}}^{2}=\sigma^{2} \in$ $(0, \infty)$. Show that $$|\frac{2}{\sigma}\left(\sqrt{S}_{n}-\sqrt{n}\right)-\...
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1answer
59 views

Rolling a die to get a sum of at least 300

The original question: Suppose you are throwing a fair-six-sided die, find the probability that at least 80 rolls are necessary to have the sum exceed 300. My solution: Let $X_i$ be the result of ith ...
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66 views

A question about Central Limit Theorem and the calculation of a limit.

The question is the next: Using CLT to calculate $$\lim_{n \to \infty} \frac{8^n}{27^n}\sum_{k=0}^n\binom{3n}{k}\frac{1}{2^k}$$ I have started by defining a sequence $X_n$ of random variables with $...
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240 views

Calculate the following limit using the Central Limit Theorem

A question about the CLT Using Central Limit Theorem to show that $$\lim_{n \to \infty} \frac{8^n}{27^n} \sum_{k=0}^n \binom{3n}{k}\frac{1}{2^k}=0$$ I have tried to define a sequence ${X_n}$ with $...
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84 views

Does the sum of random variables sampled with/without substitution differ for large populations?

We have a population of $N$ different balls. Half the balls are red, and half the balls are blue. We perform $N$ trials. In trials $i = 1,\cdots,N$ we pick a ball $B_i$ randomly. First, we pick the ...
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Stable laws of probabilty and convergence.

I'm working on a probabilty exercise and i'm stuck at some point. I did the first 3 questions without trouble. Here is what is says : We consider $(\Omega,\mathcal{A},\mathbb{P}) $ a probability ...
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1answer
20 views

Why do we need Wald's confidence interval to estimate p in a Bernoulli distribution?

I'm studying statistics and I'm a bit confused about why Wald confidence interval is needed to estimate the p in Bernoulli distribution. Let's say, I am modeling some phenomenon with a Bernoulli ...
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31 views

Tobacco content - Central Limit Theorem

Question statement: A tobacco company claims that the amount of nicotine in one of its cigarettes is a random variable with mean 2.2 mg and standard deviation 0.3 mg. However, the average nicotine ...
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21 views

Verification of computation involving Central Limit Theorem

[Image of question][1] I have attempted the question linked below. I apologise for not being able to post an image directly in my post. [1]: https://i.stack.imgur.com/996yR.png I have attempted it and ...
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Convergence in distribution of a random variable (auto regressive/time series)?

I have a hard time finding what the following task is about, I dont even know what a rv/function like this is called. Task: Let $x_t$ be a positive generated random variable, where $x_0$ is positive ...
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27 views

Slightly confused on central limit theorem

It seems that the central limit theorem has many applications, but I just want to know an extremely simple one. If I have $n$ i.i.d. random variables, $X_1,..., X_n$, and $S=\sum_{i=1}^nX_i$, does the ...
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148 views

Limiting distribution of binary variable (Central limit theorem fails)

Suppose we have a random variable $$Y_i = i \text{ with probability } \frac{1}{i}$$ and $0$ otherwise. Here all the $Y_i$ are independent. We can redefine $X_i = Y_i -1 $ so that $E(X_i)=0$. Then the ...
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Why is the output of the sum of the products of one distribution and a set of other distributions like Gaussian Distribution?

Let $p_0$, $p_1$, ... $p_n$ be $n+1$ different distributions, $A$ a 1D matrix of shape $(n, 1)$ where $a_{i1}$ is sampled from $p_0$, $X$ be a 2D matrix of shape $(k, n)$ where $x_{ij}$ is sampled ...
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51 views

Limiting distribution CLT problem

Let $X_1, X_2, ....$ be iid RV with mean 0, variance 1, and $E(X_i^4)$ is finite, Show that the limiting distribution of $Z_n = \sqrt{n} \frac{X_1X_2 + X_3X_4 +..... + X_{2n-1}X_{2n}}{X_1^2+X_2^2 + ......
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52 views

Xi are i.n.i.d bounded rv show a central limit exists

We are asked to show if $X_i$ are independent but not i.d. and where $X_i$ are all bounded with $S_n = \sum_n X_i$ and $s^2_n = Var(S_n)$ with $s^2_n \to \inf$ then $\frac{S_n}{s_n}$ has a central ...
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22 views

Square root n-inconsistent estimator

Let $\boldsymbol{\theta}_n$ be an estimator computed on a sample of size $n > 0$, where $\boldsymbol{\theta}_0$ is its true value. I establish that $$\sqrt{n}(\boldsymbol{\theta}_n - \boldsymbol{\...
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1answer
37 views

Weak Convergence of centered and scaled sum to a non-degenerate limit implies existence of the second moments of the sequence

I'm looking at the following proposition: $\text{Let}\, X, X_1,\ldots,X_n\colon (\Omega, \mathcal{A}, \mathbb{P}) \to (\mathcal{X}, \mathcal{B}) \,\text{be independently, identically distributed ...
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20 views

Convergence in distribution of sum of two random sequence

Let $\{(x_i,y_i)\}_{i=1}^\infty$ be a sequence of iid random variables, with $E|x_i|<\infty$, $E(y_i)=0$, and $E(x_iy_i)=0$. Both $y_i$ and $x_iy_i$ have finite second moment. And let $$ a_n=\frac{...
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22 views

Finding a better estimate to binomial probabilities

A plays $9$ games with his computer. In each game, he will win, draw or lose with probabilities $\frac{1}{2}$ , $\frac{1}{3}$ and $\frac{1}{6}$ respectively.A day is “great” if he wins all of the $9$ ...
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1answer
33 views

Limiting Distribution for Squared Sum of i.i.d RVs divided by Sum of Square iid RVs

I'm facing some difficulties in determining the limiting distribution of $$\frac{\left( \sum_{i=1}^n X_i\right)^2}{\sum_{i=1}^n X_i^2}$$ as $n \to +\infty$ for a sequence of i.i.d. random variables $\...
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36 views

$\sup|F(x)-G(x)| \leq \frac{1}{4} \int_{-\pi}^{\pi} \frac{|f(t)-g(t)|}{t}\,\mathrm{d}t$

When I was learning the convergence rate of the central limit theorem, I encountered a proof problem: $$ F(x)\text{ and }G(x) \text{ are two distribution functions and } f(t)\text{ and }g(t) \text{ ...
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42 views

Central Limit Theorem when variance and mean depend on N

Is there a Central limit theorem in which the variance of the random variables depends on the sample size? For example, assume $X_1$, ..., $X_n$, $n$ independent random variables such that $E(X_i) = \...
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1answer
61 views

Lyapunov's Central Limit Theorem

Let $\{X_k\}$ are independent, $S_n = \sum_{k=1}^n X_k, D_n^2 = Var(S_n) < \infty$ (a) Show that if $\exists q >2$,s.t $$ \lim_{n \rightarrow \infty} D_n^{-q} \sum_{k=1}^n E\{ |X_k - EX_k|^q \} =...
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1answer
39 views

Approximate the probability of “bad days”.

Every day A plays $9$ games with his computer. In each game, he will win, draw or lose with probabilities $\frac{1}{2}$, $\frac{1}{3}$ and $\frac{1}{6}$ respectively. He calls a day “bad” if he wins ...
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1answer
28 views

Central limit theorem approximation of distribution

I am doing a question which asks for an approximation of the distribution of X, which is the number of successes out of 50 independent trials with probability 0.4, using the central limit theorem. ...
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1answer
46 views

How to check the sequence of independent random variables satisfy Lindeberg condition?

The sequence of independent random variables has the following distribution. $$ \alpha>0\,, \quad P(X_n=n)=P(X_n=-n)=\frac{1}{6n^{2(\alpha-1)}}\,,\quad P(X_n=0)=1-\frac{1}{3n^{2(\alpha-1)}} $$ I ...
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98 views

find the limit $e^{-n}\sum_{k=0}^{n}\frac{n^k}{k!}$

I encountered the following problems when I was learning about the central limit theorem. Prove $$ e^{-n}\sum_{k=0}^{n}\frac{n^k}{k!}\rightarrow\frac{1}{2} \tag1\\ n\rightarrow\infty $$ This result ...
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1answer
42 views

Does pdf converge to the normal pdf for the sample mean?

I am watching MIT ocw lectures by Prof. Tsitsiklis on probability (youtube link is below). My doubt is regarding a point he makes in the lecture on the Central Limit Theorem. He says: The Central ...
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40 views

Convergence in distribution of the sample geometric mean for positive random variables

I am stumped on the current problem regarding asymptotic distributions of random variables. Suppose that $X_n$ are positive, iid random variables with finite mean $\mu$ and finite variance $\sigma^2$. ...
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1answer
46 views

Asymptotic distribution of the sample variance

Consider the linear model $y_i = \beta x_i +u_i$ for $i=1,...,n$ where $(x_i,y_i)$ are i.i.d. and $E(u_i\mid x_i)=0$ while $E(x_i^4)<\infty$ and $E(u_i^4)<\infty$. Let $n$ be large. Derive the ...
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1answer
44 views

CLT for functions of random variables

Let $X$ be a random variable with mean $\mu$ and variance $\sigma^2$ and $X_i$ with $i=1,...,n$ are a sequence of observations of $X$. The central limit theorem says $Z = \sqrt{n}\frac{\bar{X} - \mu}{\...
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98 views

(Durrett : Probability : Theory and Examples 5th ed, Excercise 3.4.13 ) Lindeberg Feller Theorem

The problem is Suppose $P(X_j=j)=P(X_j=-j)=1/2j^{\beta}$ and $P(X_j=0)=1-j^{-\beta}$ where $\beta>0$. Show that if $\beta <1 $ then $S_n/n^{(3-\beta)/2}\Rightarrow c \chi$. Here is one ...
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1answer
31 views

Simple hypothesis likelihood ratio test, when random variables follow Poisson distribution.

Let $H_0 = \lambda_0$ and $H_1 = \lambda \neq \lambda_0$. Then $$-2\log \lambda(y) = -2\log\frac{L(y|\lambda_0)}{L(y|\hat{\lambda})} = 2n \left(\bar{y} \log \left(\frac{\bar{y}}{\lambda_0}\right) + \...
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1answer
94 views

To find n using CLT

A distribution with unknown mean μ has a variance equal to 1.5. Use central limit theorem to find how large a sample should be taken from the distribution in order that the probability will atleast be ...

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