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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). 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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1answer
104 views

Show $X$~Exponential distribution given $P(X > nx) = (P( X > x ))^n$

Show $X$~Exponential distribution given $P(X > nx) = (P( X > x ))^n$ Can someone give some hint ? The only thing I can think of is to write it as CDF format and say according to observation ...
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22 views

When do we use the Central limit theorem to construct a confidence interval? [closed]

When should I use the Central limit theorem to construct a confidence interval? What kind of parameter would I be "estimating" with the CLT confidence interval? Thanks for any help!
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1answer
35 views

Sum of sequence of non-identical random variables. Can they become something else than Normal distributed?

The Central Limit Theorem is well known in statistics. It states: ...
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1answer
30 views

Convergence for ${\bf p}^H {\bf D}{\bf p}$ using law of large numbers

Suppose ${\bf p} = [p_1, \dots, p_n]^T$ be a $\mathbb{C}^{n \times 1}$ vector whose elements are i.i.d zero-mean and unit variance random variable (RVs), i.e., $\mathbb{E}[|p_i|^2] = 1$. Then from law ...
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1answer
56 views

Convergence for $\frac{1}{n} {\bf p}^H {\bf D}{\bf q}$ using law of large numbers

Suppose ${\bf p} = [p_1, \dots, p_n]^T$ and ${\bf q} = [q_1, \dots, q_n]^T$ be mutually independent $\mathbb{C}^{n \times 1}$ vectors whose elements are i.i.d zero-mean and unit variance random ...
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0answers
32 views

Sequence of random variables i.i.d. with normal distribution

I'm confused with this problem. I've seen this problem and I think is wrong. But if not, anyone please give me a hand. Thanks. Let $(\mathcal{X}_{i}) $ a sequence of random variables such that $ \...
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11 views

Rate of convergence in the Laplace-deMoivre theorem

Let $X_n$ have a binomial distribution with parameters $n$ and $p=1/2$, but for convenience let's assume $X_n$ is shifted and scaled so as to have zero mean and unit variance. According to the ...
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1answer
45 views

Convergence to standard normal distribution but law of large numbers does not hold. Difficult example

Let $X_1, X_2,...$ be a sequence of random variables (not necessarily independent or identically distributed) Give an example of a sequence such that $\sum_{i=1}^n ({X_i-\mu}) \over \sqrt{n}$ ...
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1answer
15 views

Continuity corrections on modelling discrete distributions

A discrete random variable X has the distribution $U(11)$. The mean of $50$ observations of $X$ is denoted by $\bar{X}$ . Use an approximate method, which should be justified, to find $P(\bar{X} \...
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15 views

Normal distribution and sample distribution question

In a recent year, the distribution of scores of students on the ACT college entrance exam was modelled by a normal distribution with a mean of 20.9 and a standard deviation of 4.7. 1) The mean score ...
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1answer
43 views

proportion of the voters/ Central limit theorem

I want to compute the proportion of the voters p. Therefore I consider random variables $X_k$ for $k=1,...,n$: $$ X_k=\left\{\begin{array}{ll} 1, party \ is \ elected: "p" \\ 0, party \ ...
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44 views

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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26 views

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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37 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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15 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
32 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
53 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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40 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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7 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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33 views

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
29 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
53 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
63 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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47 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
86 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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0answers
41 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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40 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
77 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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22 views

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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0answers
29 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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0answers
44 views

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
90 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
24 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
56 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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0answers
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 ...
3
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1answer
97 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
37 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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0answers
120 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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0answers
31 views

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
96 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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2answers
85 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
22 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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0answers
44 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
82 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 ...
2
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1answer
38 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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0answers
35 views

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
17 views

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}=\...
3
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0answers
55 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#...