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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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Probability problem about a parking lot

We want to design a parking lot for a group of 200 apartments still under construction. It is known that for each department (from city statistics) the number of cars will be 0, 1 and 2 with ...
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14 views

Convergence on Geometric distribution [on hold]

Suppose that $Xn$ ∼ $Geo( {λ/n+λ} )$ with $n = 1,2,...$ where λ is a positive constant. Show that $Xn/n$ converges on when n → ∞, and determine the parameter of the limit distribution.
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Eating competition central limit problem

Nate is a competitive eater specializing in eating hot dogs. From past experience we know that it takes him on average $15$ seconds to consume one hot dog, with a standard deviation of $4$ seconds. In ...
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32 views

Central Limit Theorem with two random variables?

Let $X_1\dots X_{500}$ be i.i.d. random variables with expected value $2$ and variance $4.$ The random variables $Y_1\dots Y_{500}$ are independent of the $X_i$ variables and also i.i.d., but they ...
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7 views

Accuracy of Euler Monte Carlo discretization without knowing exact solution?

By using Euler Monte Carlo discretization (for a Hull-White model) we simulate $$r(t+\Delta t)=r(t)+\lambda(\theta(t)-r(t))\Delta t+\eta\sqrt{\Delta t}Z$$ with $Z\sim N(0,1)$, $\lambda$, $\eta$ ...
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13 views

Do we have to assume normality of the data, even when we conduct z-test or t-test with large samples?

I read this lecture note and found that it assumes normality of the data when we conduct z-test or t-test. I can accept that when we have small samples we have to assume normality of the data, ...
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1answer
27 views

Computing probability using poisson random variable

The number of packets arriving at a multiplexer in any one second interval is a Poisson random variable with mean $15$. Assume that the number of arrivals in nonoverlapping one second intervals is ...
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1answer
44 views

Showing $X$ is standard normal when $X$ has the same distribution as $\frac{X_1+X_2}{\sqrt{2}}$

$X, X_1, X_2$ are i.i.d random variables with $\mathbb{E}[X] = 0$ and $\mathbb{E}[X^2] = 1$. Suppose $X$ has the same distribution as $\frac{X_1+X_2}{\sqrt{2}}$. I need to show that $X$ is standard ...
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36 views

Central limit theorem and integrability

If $(Y_n)_n$ is a sequence of independent random variables and identically distributed, and if $\frac{\sum_{k=1}^nY_k}{\sqrt{n}}$ converges in distribution to a random variable Y, does this mean that $...
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5 views

Estimating the discrete random walk probability by error function

I am trying to work out the asymptotic large $t$ behavior of following function \begin{equation} f(t ) = \sum_{x = 0}^{2t} { 2t \choose t + x} p^{ t+x } (1 - p)^{ t - x} = \sum_{x = 0}^{2t} { 2t \...
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Proof of the multivariate Central Limit Theorem

Casella and Lerner's "Theory of Point Estimation" (2nd edition) provides a definition of the multivariate Central Limit Theorem, for which no proof is given. What would be its derivation?
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1answer
27 views

How to check if the central limit theorem can be applied to the data?

I have some data that I know nothing about. How can I check if Central Limit Theorem apllies to it? I've made a histogram out of it, and it seems to be close to normal distribution, but is that fact ...
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2answers
44 views

Probability problem involving two normal variables

The assembly of a machine requires two stages, that proceed consecutively and independently of one another. The first stage takes a mean time of 20 minutes with a standard deviation of 8 minutes. The ...
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1answer
50 views

For missing data problem, show that $\frac{\frac1n{\sum_{i=1}^nD_iY_i}}{\frac1n{\sum_{i=1}^nD_i}}\overset{p}\to E(Y)$.

Consider a missing data $\{(Y_i,D_i):1\le i\le n\}$. If $D_i=1$, $Y_i$ is observed; if $D_i=0$, $Y_i$ is missing. Assume that $Y\bot D$. Denot $p=E(D)$, Show that $$\frac{{\sum_{i=1}^nD_iY_i}}{{\sum_{...
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1answer
53 views

Controlling the Lyapunov condition

I have been struggling with the following exercise for quite some time now. Let $(Z_n)_{n \geq 1}$ be a sequence of independent random variables such that for $j=1,2, \ldots$ we have that for some ...
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37 views

Berry-Esseen theorem for i.i.d. truncated random variables

Let $X_1,...,X_n$ be iid positive random variables and let $Y_i$ represent the truncation of $X_i$ to $(0,b]$, with $b$ a known parameter. I'd like to use the CLT to estimate the distribution of $\...
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2answers
75 views

Central limit theorem example

I am not able to make progress with the following exercise. Assignment: The advertising board is lit by one 100W bulb. Using the central limit theorem determine the minimum number of bulbs needed ...
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38 views

Which version of the central limit theorem do we need to apply here?

Let $f\in C^3(\mathbb R)$ with $f>0$ and $g:=\ln f$. Assume $g'$ is Lipschitz continuous. Let $d\in\mathbb N$ and $X$ be a $\mathbb R^d$-valued random variable with density $$\mathbb R^d\ni x\...
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23 views

CLT and sum of gamma random variables

I am having trouble approximating the sum of gamma-distributed variables via CLR. I know via Gamma that $X=\sum_{i=1}^n X_i \\$ and $X\sim\Gamma(n\alpha,\beta) \\$ and $CLT: Z_{n}=\frac{\overline{X}-...
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1answer
72 views

What are mean and variance of $W_i$, given that $Z_n=\frac{\sum{W_i}}{\sqrt{n}\sigma}\sim N(0,1)$? [closed]

Let $$Z_n=\frac{\sum{W_i}}{\sqrt{n}\sigma}\sim N(0,1),$$ where $W_i=X_i-\mu$. What are the mean and variance of $W_i$?
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Emergence of Lognormal distribution for the concentration of chemical compounds

I'm currently reviewing the literature about lognormal distributions describing/approximating the variability of a given chemical compound across different cells/ samples etc etc. The main argument ...
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18 views

Central Limit Theorem with linear transformation?

Suppose $Z=(z_{1},...,z_{m})^T$ is m dimensional vector, each $z_{i}$ is independent identical distribution with mean 0. If we do linear transformation like: $$X = \Gamma Z$$ where $\Gamma$ is $p*m(p&...
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Central limit theorem for square integrable martingales

Let $(M_t)_{t \in \mathbb{R}^*}$ be a square-integrable martingale. I am looking for a reference for the following convergence result : $$\frac{M_t}{\sqrt{\langle M_t \rangle}} \overset{d}{\to} \xi$$ ...
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1answer
52 views

Normal approximation of MLE of Poisson distribution and confidence interval

Let $(X_1,\ldots,X_n)$ denote a random sample from a Poisson distribution with parameter $\lambda$. Maximum Likelihood Estimate of $\lambda$ is given by $\hat{\lambda} = \bar{X} = \frac{1}{n} \sum\...
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1answer
20 views

How do you use the central limit theorem to approximate the sample mean when pdf f(x)=2x

Let $X_{1},\ldots, X_{n}$ be a collection of independent identically distributed (iid) samples from a population with pdf $$ f(x) = \begin{cases} 2x,&\text{if } 0 \leq x \leq 1\\ 0,&\text{...
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2answers
51 views

A transformation $g$ such that $g(S^2)$ has asymptotic distribution depending on $\beta_2$ only

Let $X_1, X_2,\ldots,X_n$ be i.i.d. RVs with $E|X_1|^4 < \infty$. Let $\operatorname{var}(X_1) = \sigma^2$, $\beta_2 = \mu_4/\sigma^4$. (a) Using CLT for i.i.d. RVs, show that $\sqrt{n}(S^2-\...
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17 views

Equidistributed sequences satisfying the central limit theorem

For a sequence (x_n) of reals that are equidistributed modulo one (such as (prime-) multiples of irrationals, or most geometric sequences) does the (properly scaled) sum of the remainders converge to ...
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1answer
36 views

Central limit theorem for weighted average

Let $(a_i)_{i\ge1}$ be a bounded positive sequence and $X_i$ be iid random variables with mean $0$ and finite variance. Let $s_n=\frac{\sum_{i=1}^n a_i X_i}{\sqrt{\sum_{i=1}^n a_i}}$. If $a_i=1$ for ...
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1answer
32 views

Central Limit theorem with extremely skewed population

If $X$~$N(1,0.1)$, and $Y=X^n$ where $n$ is very large (e.g. 200). Due to the asymetric distribution, $E(Y)$ will be skewed very far towards the extreme upper end of the distribution, dominated by ...
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77 views

Variance of sum of $m$ dependent random variables

Let $X_1,X_2,...$ be a sequence of identically distributed and $m$-dependent random variables with $\mathbb{E}[X_i]=0$, $0<Var(X_i)<\infty$ ($m$-dependent means that each $X_i$ is independent of ...
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How to show that there is a number at which the truncated Taylor Series is exact?

So, I'm reading an "easy" proof of the Central Limit Theorem in the book Probability and Mathematical Statistics, by Sahoo, and there is a point where the moment generating function is expanded in a ...
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2answers
92 views

A question about the central limit theorem

The question is: $g:R\rightarrow R$ has at least three bounded continuous derivatives and let $X_i$ be $iid$ and in $L^2$. Prove that: i) $\sqrt{n}[g(\overline{X_n}) - g(\mu)]\xrightarrow{w} N(0,g^{...
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71 views

Tilted sum of independent random variables

Let $(X_i)_i$ be a sequence of centered i.i.d. random variables with finite variance. Is it true that $$\frac{\sum_{i=1}^{\lfloor n^{0.6} \rfloor}X_i}{\sqrt{n}}\stackrel{\mbox{a.s.}}{\longrightarrow} ...
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1answer
18 views

Is Confidence Interval taken on one Random Sample or A Sampling Distribution

I am stuck at CI. Intuitively, a confidence interval (CI) is a type of interval estimate, computed from the statistics of the observed data ( random samples), that might contain the true value (mean) ...
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39 views

Doubts regarding Central Limit Theorem

All, I am studying CLT and I had a doubt. I understand that sampling distribution has a mean same as population mean and std.dev as 6/sqrt(n) where n is sample size. Each point on X axis of ...
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1answer
60 views

How to compute $\lim_{n \to \infty}P(C_n>C_0)$?

The unit price of a certain commodity evolves randomly from day to day with a general downward drift but with an occasional upward jump when some unforeseen event excites the markets. Long term ...
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1answer
32 views

Convergence (distribution)

$X_1, X_2, X_3....$ are independent random variables. $P(X_n=0)=P(X_n=2)=1/4, P(X_n=-1)=1/2$. Find the limit of: $\frac{4\sqrt{n}(X_1+X_2+...+X_n)-7n}{n+(X_1+X_2+....+X_n)^2}$. I computed: $EX_n=...
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Central Limit Theorem- Lapunow, Linderberg

I have a task: $(X_n)_{n>=1}$ are independent. $P(X_n=0)=1/n$ and $P(X_n=2n)=1-1/n$. Check the weak convergence $\frac{X_1+X_2+X_3+....+X_n}{n}-n$. I tried use the Lapunow theorem or Linderberg ...
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1answer
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Problem with Linderberg condition

I am trying to prove that $$\frac{S_n=\sum_{k=1}^n X_k^2-n}{\sqrt{n}}$$ converges in distribution, where $X_i$ are iid. and are normally distributed with mean 0 and variance 1. I defined $X_{n,k}=\...
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51 views

How to handle little $o$ in the central limit theorem

I am having some trouble understanding a couple of lines in the proof of the central limit theorem using characteristic functions: https://en.wikipedia.org/wiki/Central_limit_theorem#...
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1answer
27 views

Lindeberg CLT condition on Discrete Uniform independent sequence of random variables

There is a worked exercise in my book, however there is a line that I am not sure sure about. I understand all of the work before and after this line to finish the proof. Here is what we are given: ...
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1answer
41 views

Proving IID Central Limit Theorem using Lindeberg Conditions.

The goal is to prove the IID Central Limit Theorem through Lindeberg's Condition. Suppose that $X_1,X_2,\ldots\displaystyle\sim\text{i.i.d.}$ with $E[X_i]=\mu$ and $Var[X_i]=\sigma^2<\infty$. ...
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1answer
19 views

Trimmed second moment going to zero by LDCT

We have $0<\sigma<\infty$ and $\epsilon >0$ and $X_1, X_2,...$ iid. The argument involves CLT and it continues on until the line below: That $\frac{1}{\sigma^2}E\big[(X_1-\mu)^2\mathbb{1}\...
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1answer
24 views

limiting distribution of Xbar considering an autoregressive process

Here is the question: Consider the stationary Gaussian autoregressive process of order 1, $X_{i+1} − μ = ρ(X_i − μ) + \sigma Z_i$, where $Z_i$ are iid N(0, 1). Find the limiting distribution of $\...
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1answer
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A die is thrown until the first time the total sum of the face values of the die is 700 or greater. What is the probability for $n$ tosses?

Estimate the probability that, for this to happen, more than 210 tosses are required less than 190 tosses are required between 180 and 210 tosses, inclusive, are required We're supposed to be using ...
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2answers
46 views

Prove Rate of Convergence of Monte Carlo

Let $X_1, X_2, \ldots$ be i.i.d. random variables with mean $\mu$ and variance $\sigma^2$. How does \begin{equation} \mathbb E\left[\,\left|\frac{1}{N} \sum_{i=1}^n X_i - \mu\, \right|\,\right] \to O\...
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1answer
28 views

Limit of a Symmetric Random Walk

I'm given a probability space of ($\Omega$, $\mathcal{F}$, $\mathbb{P}$) and am asked to look into a symmetric random walk with its n-step defined as $$ X_k = \Bigg\{ \begin{matrix} 1 & \text{...
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35 views

central limit theorem: what is the variance?

This is a very basic question that I'm pretty sure I understand but I wanted to double check. Given a regression model of: $$y_t = \mathbf{x_{t}^{\prime}}\beta + u_t$$ We can use one of the CLT ...
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0answers
110 views

Prove the central limit theorem for a sequence of i.i.d. Bernoulli($p$) random variables

Prove the central limit theorem for a sequence of i.i.d. Bernoulli($p$) random variables, where $p\in(0,1)$. I am trying to do this by computing the moment generating function of the object I want ...
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1answer
35 views

Generating a number belonging to N(0,1) using *m* numbers from U(0,1) using central limit theorem

I was going through a blog which details how to generate a multivariate Gaussian vector, given a mean vector μ and co-variance matrix σ. As a starting point, author uses generated uniform ...