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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20 views

Sum of independent but not identically distributed uniform random variables

Let $(X_{j})_{j\geq1}$ be independent and uniformly distributed on $(-j,j)$ and let $S_{n} = X_{1} + ... + X_{n}$. Show that $\lim_{n \to \infty}S_{n}/n^{3/2}=Z$ in distribution where $Z$ ~ $N(0,1/9)$ ...
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29 views

When can the level of the test be exactly $\alpha ?$ in non randomized test. And how to use CLT to find the critical value.

Let $X_{1}, \ldots, X_{n}$ be a sample from the Bernoulli distribution with parameter $p$ Consider testing $H_{0}: p=p_{0}$ versus $H_{1}: p=p_{1}$ where $p_{0}<p_{1}$ are known numbers. (a) Using ...
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33 views

How to determine confidence interval using central limit theorem?

How to determine the confidence interval for the unknown theta parameter of the Uniform($[0, \theta]$)-distribution using the central limit theorem, considering that the significance level is given ...
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14 views

asymptotic normality of z-estimator

I'm working on Problem 5.4.1 in Bickel and Docksum's Mathematical Statistics Let $X_1, \dots, X_n$ be i.i.d. random variables distributed according to $P\in\mathcal{P}$. Suppose $\psi:\mathbb{R}\to\...
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16 views

Let $X_1,X_2,\dots$ be i.i.d. and set $N_n(a,b)=\sum_{k=1}^n\mathbf1_{\{c+a/n<X_k<c+b/n\}}.$ Then $N_n(a,b)$ converges in distribution

Problem: Let $X_1,X_2,\dots$ be i.i.d. random variables with distribution function $F$ and a continuous density function $f$. Let $c\in\mathbb R$ be a number with $f(c)>0$ and consider $$N_n(a,b)=\...
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23 views

Does uniformly bounded second moment imply Lindeberg's condition>

Suppose an independent sequence $X_i \in \mathbb{R}^k$ is such that $\sup_n E(\Vert X_i \Vert^2) \leq M$ for some constant $M$. Does this imply that $X_n$ satisfies the following Lindeberg-type ...
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28 views

Why does the form of the normal distribution have different forms for the 2nd parameter?

Sometimes I see the normal distribution written as $N(\mu,\sigma)$ sometimes as $N(\mu,\sigma^2)$ and sometimes, when the Central Limit Theorem is involved, as $N(\mu,\sigma^2/n)$ As a software ...
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19 views

Convergence of chi-squared distribution to standard normal

I want to prove that $\frac{(X_n-n)}{\sqrt{2n}}\xrightarrow{{L}}Z\sim N(0,1)$ as $n\rightarrow\infty$ where $X_n\sim\chi^2_n$. I don't want to use the CLL directly. So far this is what I got, but I am ...
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Stronger convergence in the Central Limit Theorem

In the proof I have of the CLT theorem, we use the fact that, for $(X_i)_{i\geqslant 1}$ a sequence of iid random variables in $L^2(\mathbb{P})$, we have $$\lim\limits_{n\to+\infty}\Phi_{Z_n}(u)=\exp\...
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11 views

Hypothesis testing Z-test:Where does the normal curve come from?

When performing a hypothesis testing z-test we can in the end determine the p-value using a standard normal curve after determine the z-score. Is the standard normal curve derived from my sample, that ...
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21 views

How to find sample mean

The standard deviation of the starter salary for economics students graduation is 4200. Calculate the probability that sample mean of 40 student(x.bar) will be bigger than mean of the total population ...
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21 views

Limiting Value Sum $c^X$ for Uniform $RV$

I have $n$ draws $X_i$ from a Uniform r.v. on $[0,1]$ and want to find the limit of $\frac{1}{n} \sum_{i=1}^n c^{X_i}$ where $c$ is a known constant in $[0,1]$. My thought is to transform this ...
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45 views

Connection between Gauss and Poisson distribution?

I know that these distributions are connected by the central limit theorem. But as it is written here: The central limit theorem says that the distribution of the mean of N draws from a probability ...
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31 views

A question about part of the proof of the Central Limit Theorem

I've been reading the proof the Central limit theorem from Rick Durret's book. In Theorem 3.3.8 he proves the following If $E[X^2]<\infty $ then $$\tag{1}\phi(t)=1+itE[X]-\frac{t^2}{2}E[X^2]+o(t^2)...
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159 views

Poisson limit process with divergence condition

Please help me with problem 3.26 of 'Probability Theory' by Varadhan: For each $n\in \mathbb{N}$, $\ X_{n,j}:(\Omega,P) \to \mathbb{R}$ with $1\leq j\leq k_n$ are $k_n$ independent random variables ...
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Simple algebra issue with regards to finding the correct $Z$ value when using the Central Limit Theorem

Suppose that the number of traffic accidents, $N$, in a given period of time is distributed as a Poisson random variable with $E(N) = 100$. Use the normal approximation to the Poisson to find $\Delta$ ...
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36 views

CLT for independent, but non-identically distributed Poisson random variables

Show that $$\frac{\sum_{i=1}^n(Y_i-\lambda_i)}{\sqrt{\sum_{i=1}^n\lambda_i}}\overset{d}{\rightarrow}\mathcal N(0,1)$$ where $\{Y_i\}\sim\mathsf{Pois}(\lambda_i)$ are independent and $\sum_{i=1}^n\...
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28 views

CLT for non identically distributed bounded random variables

Let X1, X2, ... be independent, zero-mean, bounded random variables. Let $S_n$ = $X_1 +... + X_n$ with variance $s^2_n$ = Var($S_n$), $s^2_n \rightarrow \infty$. Show that $S_n/s_n$ has a central ...
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Showing two versions of the central limit theorem are equivalent (are they?)

I am trying to read Norris' book on Markov chains. In particular, at page 160 he states the central limit theorem : Theorem 4.4.1 (Central limit theorem). Let $X_1, X_2,...$ be a sequence of ...
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35 views

Expectation of Gaussian CDF

I have $X_i$ from $n$ i.i.d draws of N($\theta$,1). By the SLLN, I know $\bar{X_n}$ converges almost surely to $\theta$. I would like to find the limiting distribution of $\phi(x-\bar{X_n})$ for fixed ...
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Central Limit Theorem with Volume

Suppose are growing limes. Suppose the mean radius of the limes is 9cm with standard deviation 2 cm. Suppose a bar who will buy our limes expects the volume of the limes to be between 8π cm3 and 12π ...
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70 views

Variations on Gauss' integral trick

This question is inspired by these two: Non-trivial values of error function erf(x)? Where is the mass of a hypercube? Upon reading these two, I realized there might be a geometric way to compute ...
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24 views

How does the central limit theorem enable the approximation of probabilities?

My text book states the following about the Central Limit Theorem and then launches into examples. Let $X_1, X_2 .... X_n$ be a sequence of independent and identically distributed random variables ...
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38 views

Central Limit Theorem for difference of two sample means

According to Walpole's Probability and Statistics for Scientists and Engineers, the "central limit theorem can be easily extended to the two-sample, two-population case." That is, if ...
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36 views

Application of the central limit theorem on Markov chains

I'm studying a paper from Rosenfeld about bitcoin mining pools. He formulates a Markov chain by the definition: \begin{equation*} \begin{aligned} X_{t+1} - X_t= \begin{cases} B - (1 - f) \...
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31 views

converge in distribution imply converge of moments for ${x_i}$ that satiefies the Lyapunov condition?

if $\{x_i\}$ is a sequence of zero mean independent random variables that satisfies the Lyapunov condition:$$s_n^{-2r}\sum_{i=1}^nEx_i^{2r} \to 0$$ as $n \to \infty$. here $s_n^2=\sum_{i=1}^nEx_i^2$, ...
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10 views

Local CLT of the simple random walk on $\mathbb{Z}^d$

Consider a simple random walk $S=(S_0,S_1,...)$ on $\mathbb{Z}^d$ and let $p_n(x) = \mathbb{P}_0[S_n =x]=p^n(0,x)$. If $x\in \mathbb{Z}^d$ has the same parity as $n\in \mathbb{N}$, i.e., $n+x_1+\cdots+...
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Probability distribution of sample mean estimator in case of Latin-hypercube sampling

The sample mean estimator for a function $g(\, \cdot \,)$ of a random variable $Y$, for $N$ independent realizations of $Y$, i.e., $$T\left( {{Y_1}, \cdots {Y_N}} \right) = \frac{1}{N}\sum\limits_{i = ...
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43 views

Probability of getting within $k$ heads of $50$

You flip a coin $100$ times. What's the probability that you get between $[50 - k, 50 + k]$ heads for some integer $k$? I know we can use the CLT with mean $50$ and standard deviation $5$, and I end ...
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10 views

Central limit theorem in hierarchical model

This is an exercise in 《Statistical Inference Second Editition》by George Casella Roger L. Berger. Since this is an exercise, I just want some hints. Given that $N=n,$ the conditional distribution of $...
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58 views

Convergence in probability from the central limit theorem

My question comes from the proof of the Delta method. One of the conditions states that $\sqrt{n}(Y_n - \theta) \rightarrow N(0,\sigma^2)$ in distribution for some sequence of random variables $Y_n$. ...
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What is a specific observation of a probability parameter estimate? Programmer trying to understand statistics

I am trying to understand the following statement about a collection of independent and identically distributed Bernoulli random variables. We have $\theta$ as the probability of success of the ...
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42 views

Convergence Rate of Sample Variance

Question: Given $X_1,X_2,...$ as an sequence of iid distributed random variables with $E(X_i)=0 $ and $V(X_i)=σ^2$ and the fourth order moment $E(X_i^4)<\infty$. Show that: $\sqrt{n}(S_n^2-\sigma^2)...
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Confused by subscript of Z Score in EBM notation

At openstax.org I see $$ EBM = (Z_{\frac{\alpha}{2}}) (\frac{\sigma}{\sqrt{n}}) $$ However on the next line $Z_{\alpha}$ is used instead. Sometimes I see that Z refers to the area on the left of the Z ...
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Whether my understanding of the Central Limit Theorem is a correct way to look at the idea? + a few small questions

For the last while I've been attempting to truly understand what is being said by the Central Limit Theorem. I get the general idea, but there are still one or two details that are troubling. To make ...
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41 views

Why do we sum the sample variance when shoveling snow?

My text book has the following question A highway department has enough salt to handle a total of 80 inches of snowfall. Suppose the daily amount of snow has a mean of 1.5 inches and a standard ...
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48 views

Can a normal distribution be approximated with a binomial distribution?

We know that a binomial distribution can be approximated with a normal distribution when $n * p$ is large where $n$ and $p$ are the number of bernoulli trials and $p$ is the probability of success of ...
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22 views

Using Central limit theorem for an equation

Suppose $X_1, X_2, ..., X_n$ are sequences of iid random variables with mean equal to zero and variance = $σ^ 2$. we define $Y_n = \frac{s_n}{σ∗\sqrt{n}} −\frac{s_{2n}}{σ∗\sqrt{2n}}$ , $S_n = X_1 + ...
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26 views

Central limit theorem for proving equation

Suppose $X_1, X_2, ..., X_n$ are sequences of iid random variables with mean equal to zero and variance = $σ^ 2$. we define $Y_n = \frac{s_n}{σ∗\sqrt{n}} −\frac{s_{2n}}{σ∗\sqrt{2n}}$ , $S_n = X_1 + ...
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23 views

Central Limit Theorem - Clarification on notation of sums and sample mean

I am having trouble completely reconciling the central limit theorem in its various forms. Specifically it has to do with notation. If we have a sequence $X_{1}, X_{2}, \dots$ of i.i.d random ...
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29 views

Why would sample probability more than zero equal sample probability more than half?

In calculating probabilities for Roulette, my text book asks me to calculate the probability of winning after 1,000 bets. Where spending \$1 has a $\frac{1}{38}$ probability of winning \$35 and a $\...
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24 views

Limit of probability that sum of $n$ iid variables is greater than $0$

How can I find the probability $$\lim_{n \to \infty} P\left(\sum_{i=1}^n X_i>0\right) = \lim_{n \to \infty} P\left(\frac{1}{n}\sum_{i=1}^n X_i>0\right)$$ if $X_i$ are iid? If the expectation of $...
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CLT for weighted sum of Bernoulli Variables

Suppose I have the random variable $y_i = w_i x_i z_i$, with $z_i$ beeing a Bernoulli variable, $x_i$ is drawn from a unimodal distribution with finite variance, and $w_i$ are constants in some ...
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50 views

Sums of binomial coefficients - accuracy of CLT approximation?

I was trying to figure out the asymptotics of the sum $$S(n) = \sum_{k=0}^{\lfloor \alpha n \rfloor} \binom{n}{k}$$ for $0 < \alpha < 1/2$ (actually, the specific case of interest for me is $\...
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60 views

Central Limit Theorem - Wikipedia article

I am trying to follow the CLT proof of Wikipedia's article on CLT I can follow up to and including the part that $Z_{n} \to N(0,1)$ as $n \to \infty$. I cannot understand the very last sentence of ...
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Do the reciprocals of asymptotically normal random variables fall into the domain of attraction of the Cauchy law?

According to the generalized central limit theorem we know that the reciprocal of a normal random variables fall into the domain of attraction of the Cauchy law, that is, for $X\sim\mathcal N(\mu,\...
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49 views

CLT and Convolutions on a diverging standard deviation

Suppose we have a probability distribution p(x), and we wish to analyze its convolution to itself n times as n approaches infinity. https://en.wikipedia.org/wiki/Convolution_power and several other ...
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36 views

a form of central limit theorem of a markov chain

Could anyone explain to me what it means by the central limit theorem for the following Markov chain? $X_{n+1}=f_{\omega_n}(X_{n}),\quad n=0,1,2,\dots$, where $\omega_0, \omega_1\dots$ are i.i.d ...
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69 views

Central Limit Theorem and Strong Law of Large Numbers. Proof that converges in distribution to $N(0, e^2)$

I have this exercise of my homework. Let $\{ X_n: n \geq 1\}$ be independent random variables and identically distribuited with uniform distribution $U(0,1)$ and $Y_n=(\prod_{i=1}^n X_i)^{-1/n} $. ...
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33 views

How to prove that an exponent comes outside of the average of an exponential? (central limit theorem moment-generating function proof)

Let us define the moment-generating function $\Gamma$ below: $$\Gamma(\lambda) = \langle e^{\lambda x}\rangle = \int_{-\infty}^\infty dx \, e^{\lambda x} P(x)$$ where the angle brackets in $\langle e^{...

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