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).

2
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0answers
28 views

Is Convergence of Moments true in Central Limit Theorem?

My question is simple. Suppose that $X_1,X_2,\ldots$ is an infinite sequence of Rademacher random variables (i.e. $\mathbb{P}(X_1 = -1) = \mathbb{P}(X_1 = 1) = 1/2$). Let $S_n$ denote the random walk $...
2
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0answers
29 views

Convergence in distribution of random sums of random variables

I have been working on the following exercise: Let $\{X_{k}\}_{k\geq 1}$ be i.i.d. random variables and let $\{\xi_{n}\}_{n\geq 1}$ be Poisson random variables with $\xi_{n}\in Po(n)$. Assume further ...
0
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0answers
24 views

Normally distributed experiment: argue that this is (is not) true

I have defined a set of graphs, where each graph of the set fulfill the following conditions has $n$ nodes degree is fixed to $d$ The edges are distributed in a way such that there are no bridges (...
0
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0answers
19 views

Probability space of random variable according to central limit theorem

According to Central Limit Theorem: $\Omega$ = {$\omega_1$, $\omega_2$,..., $\omega_n$}, $Event$, $p$, $X$ : $\Omega$ $\rightarrow$ ${\rm I\!R}$ $n$ elements $e_1$,...,$e_n$ are drawn ...
0
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1answer
27 views

Central limit theorem/ poisson distribution

Let$\ X_1,X_2,...,X_n$ be independent Poisson random variables with parameter$\ λ=1$, use the Central Limit Theorem to prove: $\ \lim_{n→∞} \frac{1}{e^n} \sum_{k=0}^n \frac{n^k}{k!} =\frac{1}{2}$ My ...
0
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0answers
34 views

Approximation error from using the Central Limit Theorem

A bank accepts rolls of pennies and gives 50 cents credit to a customer without counting the contents. Assume that a roll contains 49 pennies 30 percent of the time, 50 pennies 60 percent of the ...
1
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0answers
19 views

Any central limit theorem-like theorem stating distribution of something will be convergent to uniform distribution?

The central Limit Theorem states that the distribution of the sum of independent random variables (sampled from any population) converges to a normal distribution. Are there any similar theorems for ...
1
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0answers
13 views

Central Limit Theorem and Law of Large Numbers for stable linear stochastic systems

I am new to Markov chain in continuous state space. It would be great if anyone can help me with this. Consider a linear stochastic system: $x_{t+1}=Ax_t +w_t$, where $w_t$ i.i.d. Gaussian with zero ...
0
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1answer
37 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 ...
1
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0answers
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!
0
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1answer
29 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: ...
0
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1answer
28 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 ...
1
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1answer
53 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 ...
0
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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 $ \...
0
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0answers
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 ...
1
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1answer
42 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 $\sum_{i=1}^n ({X_i-\mu}) \over \sqrt{n}$ ...
1
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1answer
13 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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0answers
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 ...
0
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1answer
41 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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0answers
40 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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0answers
18 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 ...
0
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0answers
33 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 ...
0
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0answers
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 $Z\sim N(0,1)$, $\lambda$, $\eta$ ...
0
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0answers
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, ...
2
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1answer
29 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 ...
0
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1answer
45 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 ...
1
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0answers
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 $...
0
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0answers
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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0answers
19 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?
0
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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 ...
0
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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 ...
0
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1answer
52 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_{...
0
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1answer
58 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 ...
1
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0answers
42 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
77 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
39 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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0answers
25 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}-...
-1
votes
1answer
74 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 $W_i$?
3
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0answers
17 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 ...
1
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0answers
23 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&...
3
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0answers
32 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 : $$\frac{M_t}{\sqrt{\langle M_t \rangle}} \overset{d}{\to} \xi$$ ...
0
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1answer
66 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\...
0
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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{...
1
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2answers
52 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-\...
0
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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 ...
2
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1answer
42 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 ...
1
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1answer
33 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
97 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 ...
0
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0answers
26 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 ...
3
votes
2answers
93 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^{...