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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Using Central Limit Theorem to estimate deviation of mean estimator of sampled Bernoulli r.v.
I am answering the below question from recitation 20, this course.
The question is:
"In your summer internship, you are working for the world’s largest producer of lightbulbs. Your
manager asks ...
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1
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Laplace distribution CLT
I'm really stuck with this problem that my statistics professor gave me:
n = 81 measurements X = (X1,…,X81) were made according to a Laplace(θ),
$f_{\theta}(x) = \frac{1}{2} \theta e^{-\theta|x|} $.
...
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Can Central Limit Theorem say anything about Probability Density Function?
For a sequence $\left( X_1, X_2, \dots \right)$ of independent and identically distributed random variables with zero expectation and unit deviation, CLT implies
$$
\lim_{n\to\infty} P\left( \frac{1}{\...
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Donsker's theorem for multivariate Brownian motion [closed]
Let $B = (B^1, \dots, B^n)$ be an $n$-dimensional Brownian motion (i.e. $B = (B_t)_{t \geq 0} \in \Omega \rightarrow (\mathbb{R}^n)^{[0,\infty)})$.
Do we have something as Donsker's theorem to show ...
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Reverse of the Central Limit Theorem
I have a question about a statistical concept and would like to gather your thoughts on it. As we know, the Central Limit Theorem states that under certain conditions, the sum or average of ...
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$Z_{n,m} = \frac {(X_n +Y_m) -(n+m)}{\sqrt{X_n+Y_m}}$ converges to $N(0,1)$
Given $X_n \sim P(n), Y_m \sim P(m)$ independent. How can I see that $Z_{n,m} = \frac {(X_n +Y_m) -(n+m)}{\sqrt{X_n+Y_m}}$ converges to $N(0,1)$ when $n,m$ go to inifinity? I know that $Z_{n,m} \sim P(...
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Prove that Brownian Motions have normal distribution using central limit theorem
In the book Brownian Motion, 3rd edition by Rene Schilling, he defines a $d$-dimensional Brownian motion $B = (B_t)_{t\geq0}$ indexed by $[0,\infty)$ taking values in $\mathbb R^d$ as a process that ...
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A doubt concerning the central limit theorem and how it is presented in explanatory videos
As a mathematician currently pursuing a master's degree in machine learning, I aim to develop an intuitive understanding of statistics, particularly concerning the Central Limit Theorem (CLT). While I ...
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Who needs the law of large numbers when you have the central limit theorem
The central limit theorem (CLT) and law of large numbers (LLN) are both statements about the sample mean. If the sample mean of $n$ samples is $\bar{X}_n$, the CLT says that the distribution of:
$$\...
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"It is no restriction to assume, w/o loss of generality, that"
In this proof for CLT, the author said "it is no restriction to assume that $\mu=0, \sigma=1$". This is not clear to me. Can anyone share your thoughts?
My attempt
Note that
\begin{...
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Is there a CLT-type result for $S^2$?
The subject line is essentially the question: it there a theorem, like the CLT for the sample average, that allows us to say anything useful about the distribution of the sample variance as the sample ...
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Proving Convergence to the Lower Bound and Upper Bound for Monotonically Decreasing and Increasing Sequences
Let, a sequence $a_n$, where Minimum value of $a_n=b$ and Maximum value of $a_n=c$.
If $a_{k+1}<a_k$, Then, it has a limit $L_1=b$
If $a_{k+1}>a_k$, Then, it has a limit $L_2=c$
It is quite ...
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How to arrive at well-know result of CLT from specific version of CLT?
The version of the Central Limit Theorem I learned in class follows:
Given ${X}_{i},..., {X}_{n}$ iid RVs, such that $E[{X}_{1}] = 0 < \infty$ and $Var({X}_{1}) = {\sigma}^{2} < \infty,$ $\dfrac{...
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Show that if $\frac{S_n - m_n}{s_n} \xrightarrow{d} \mathcal{N}(0, 1)$ with $X_k \sim \text{Ber}(p_k)$ then $\sum_k p_k(1-p_k) = +\infty$
Let $(X_n)$ be a sequence of independent random variables, $X_k \sim \text{Ber}(p_k) \ \forall k \ge 1$. Set $S_n = \sum_{k = 1}^n X_k, m_n = \sum_{k = 1}^n p_k, s_n^2 = \sum_{k = 1}^n p_k(1-p_k)$. ...
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Bound on error when Poisson distribution is approximated by normal distribution
At several resources I found the claim that for large $n$ the Poisson distribution with mean $n$ is approximately normal with mean and variance $n$. Quite a few of these resources quote the central ...
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Applying Central Limit Theorem to an exponential distribution - how big should sample size be?
For an exponential distribution, in order for the sampling distribution of its mean to be well approximated by normal distribution (via central limit theorem), how big should a "typical" ...
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Prove divergence of a series with the Central Limit Theorem
In Probability Theory by A. Klenke (3rd version), in the proof of Theorem 19.35 it is required to show that almost surely:
$$
R_W^+:=\sum_{n=0}^{\infty}\exp\big(\sum_{k=0}^{n}X_k\big)=\infty
\\
R_W^-:=...
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Solving a limit using probability theory.
I’m stuck on the following probability question:
Calculate the limit
\begin{aligned}\cdot \\
\lim _{n\rightarrow \infty }\sum ^{n^{2}+3n}_{k=n^{2}+2n+1}e^{-n^{2}}\dfrac{n^{2k}}{k!}\end{aligned}
One ...
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Why do I not get a standard normal random variable after applying central limit theorem?
Suppose that $S_n = \sum_{i=1}^n X_i$, where $X_i$ has $ P(X_i = 1) = P(X_i = -1 ) = 1/2$.
I am trying to use central limit theorem to calculate $$\lim_{n \to \infty} P\left(S_n < \frac{1}{3} \sqrt{...
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Can weak convergence imply orders of mean and variance? [closed]
Suppose we have the following weak convergence:
$$
n^{-1/2}\cdot (X_n-n) \Rightarrow X \quad \text{as} \quad n\to\infty,
$$
where $X_n$ and $X$ are random variables (note: $X$ is a non-degenerate r.v.)...
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Is Lindeberg theorem error = o(1)?
from Lindeberg_Feller theorem { Theorem 4.12 in Foundations of Modern Probability by Olav Kallenberg}
let $\xi_{n,i}$ be a triangular array of row wise independent random variables with $E[\xi_{n,i}]=...
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How CLT states the sampling distribution of one large sample while sampling distribution needs multiple samples?
I have a bit confusion of understanding. The central limit theorem in statistics states that, given a sufficiently large sample size, the sampling distribution of the mean for a variable will ...
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Help understanding CLM equation "If x∼N(μ,σ) then x¯∼N(μ,σn‾√)" [closed]
Stats n00b. What does N(μ, σ) mean?
I'm trying to piece it all together, but can't fill in all the gaps (search engines don't recognise it and I can't seem to find anywhere that uses this same formula ...
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Limit central theorem - binominal distribution inequality ( Independant binominal Random variables )
$$\lim _{n\to \:\infty }\mathbb{P}\left(\sum _{n=1}^{\infty }\:X_n>\frac{pn^2}{2}\right)=?$$
When $p\in (0,1)$ , $\left\{X_n\right\}^{\infty }_{_{_{n=1}}}$, $X_n~Bin\left(n,p\right)$, as they are ...
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central limit theorem for random field
Let $x\in(1,\infty)$ and let $Z(x)=\sum_{i\ge1}Z_i(x)$ be a sum of independent random variables $Z_1(x),Z_2(x),\dotsb$ such that each random variable is bounded as $Z_i(x)\in[-1/i,1/i]$. Moreover, it ...
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Little-$o$ notation in CLT proof using Levy Continuity Theorem
I am stuck at understanding the little-$o$ notation used in the proof of CLT using Levy's theorem. Specifically, I want to understand why for fixed $n$,
$$
1 - \frac{t^2}{2n} + o\left( \frac{t^2}{n} \...
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Central Limit Theorem with two iid
Suppose$\left \{ X_1,\dots,X_n \right \}$ is a sequence of independent and identically distributed(i.i.d) random variables
with finite expected value $E[X_i]=\mu _{x}$ and variance $Var[X_i]=\...
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Central Limit Theorem Variant [closed]
Given an input of 4096-dimensional independent and identically distributed (iid) vectors, and an output obtained by performing weighted summation on these iid vectors using coefficients that are also ...
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Local limit theorem for not-identically distributed series?
I'm wondering if there is a local limit theorem for sums of independent random variables $X_1+X_2+\dots+X_n$ which are independent but not identically distributed? I know some form of CLT holds with ...
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One of the Specific Intuitions behind Central Limit Theorem
This is a relatively vague question since it occurred during the class couple of semesters back when the Professor tried to explain the "intuition" of Gaussian distribution. The information ...
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Comparing Confidence Given by Concentration Inequalities and Central Limit Theorem
Given an i.i.d. sample $X_1,\dots, X_n$ from Ber($p$), by Chebyshev's inequality,we have $$\text{Pr}(|\overline{X}-p|\le \epsilon)\ge 1-\frac{p(1-p)}{n\epsilon^2}.$$
Therefore, if $\text{Pr}(|\...
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Are observations random variables?
I'm looking at https://en.wikipedia.org/wiki/Central_limit_theorem and more specifically, at this part. When it says 'Let $X_1, X_2, \dots, X_n$ denote a random sample of $n$ independent observations',...
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Suppose that $X_1, \cdots, X_n$ is a Uniform$[0,1]$ data sample. Find the large-sample limiting distribution of $\sqrt{n}(\bar{X}/S - \sqrt{3})$
Suppose that $X_1, \cdots, X_n$ is a Uniform$[0,1]$ data sample, for large $n$.
By the Central Limit Theorem we know that $\sqrt{n}(\bar{X} - 1/2) \rightarrow^D N(0, 1/12)$.
If $E[X^4] \lt \infty$, we ...
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Distribution of the mean of samples taken from two different distributions
Consider you have some distributions $Z_1$ and $Z_2$ of which you select $n_1$ samples from $Z_1$ and $n_2$ samples from $Z_2$.
We now add up these samples and take the mean. The question becomes, ...
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Probability that the number of dice rolls exceeds 210? CLT applicable?
I can't seem to crack this nut and it would be great if someone has a hint/solution. I think a clever application of the Central Limit theorem is needed here, but I don't know how.
Suppose I roll a 6-...
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Can I use assume a Normal Distribution to calculate a Confidence Interval for multiple random variables with different means and std each?
Let's say, I've a table with a list of trips I need to make this week
and I want to calculate a confidence interval for the average trip duration time
for all the trips I'll make in the week.
Var
...
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Central Limit Theorem and Law of Large Numbers for Non-Constant "N"?
This is a question I have been having for a while.
Usually, we define the Central Limit Theorem as:
Let $X_1, X_2, \dots, X_n$ be a random sample of size $n$ from a population with mean $\mu$ and ...
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Central limit theorem and convolving a function with itself infinite times
There is something I don't understand about the central limit theorem in the frequancy domain.
The central limit theorem tells us that if we have $n$ probability density functions $f_i$ then
$$h=f_1*\...
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On the distinction between “Pairwise independent” and “Mutually independent” random variables
I have familiar with the fact that for $N$ events, we have the concepts of pairwise independent and mutually independent. From this I am interested in extensions of this notion to random variables. ...
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How does the Radial Basis Functions transform multimodal distributions to normal distributions?
In the book Hands-On Machine Learning by Aurélien Géron, in chapter two, the author states:
Another approach to transforming multimodal distributions is to add a feature for each of the modes (at ...
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Locating variable based on confidence interval for exponential distribution
$X$ is a random variable and its PDF is:
$$
f(x;θ) = \begin{cases} θe^{-θx}, & x > 0 \\\ 0, & x \le 0 \end{cases}
$$
Now, I would like to find the confidence interval regarding parameter $...
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Central limit theorem for uncorrelated and identically distributed random variables
I'm trying to determine whether a sum of identically distributed but only uncorrelated continous random variables could possibly converge to a normal distribution. First of all we define the i.i.d. ...
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Does Hoeffding's bound give tighter bound than Central Limit Theorem does on sample dependency?
From Central Limit Theorem, we know that $\mu - \bar{x}$ has a variance of $\frac{\sigma^2}{n}$ where $\sigma^2$ is the population variance. We can roughly interpret it is as $P(|\mu - \bar{x}|\ge\...
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Uniform convergence in distribution implies convergence of moments
I am reading a paper in which the author wants to prove the convergence of the moments. He transforms the object of interest $\varepsilon^{-1} (\vartheta_\varepsilon^*-\vartheta_0)$ into
\begin{align*}...
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An intuitive derivation of a central limit theorem for renewal processes
I am reading "Introduction to Stochastic Processes" by Lawler, and I am having trouble understanding his intuitive explanation of a CLT for renewal processes (Chapter 6, p.134).
Here is his ...
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CLT Problem of $N_t=\sup\{n\geq 1 | S_n \leq t\}$
Let $(Y_k)$ be i.i.d denote $\mu =\mathbb{E}[Y_1],\ \sigma ^2 = Var[Y_1]$ and there exists $c>0$ s.t $P(Y_1\geq c)=1$
$$ S_n=Y_1+...+Y_n,\ \ \ N_t=\sup\{n\geq 1 | S_n \leq t\}$$
for $t>0$
a) ...
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$\frac{X_1+...+X_{n-1}-\log n }{\sqrt{\log n} }\rightarrow N(0,1)$ where $X_n\sim \operatorname{Ber}(\log \frac{n+1}{n})$
Prove : $\frac{X_1+...+X_{n-1}-\log n }{\sqrt{\log n} }\rightarrow N(0,1)$ where $X_n\sim \operatorname{Ber}({\lambda}_n)$ and $\lambda _n = \log \frac{n+1}{n}$
I tried using a similar proof to CLT ...
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Central Limit Theorem with continuously differentiable function.
I have to prove this variation of the Central Limit Theorem:
Let $(Z_n)_{n \in \mathbb{N}}$ be i.i.d. random variables with existing variance $\sigma^2 = \mathrm{Var}(Z_1) > 0$ and $\mu = \mathrm{E}...
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Using the central limit theorem on a binomial random variable
I need to show that the limit as n goes to infinity of $P(X_n \geq \frac{n}{2})$ is 1/2 if the expected value is equal to 1. The random variables are binomially partitioned but it's not given that ...
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Some hint on Generalization or Extension of the convergence related to random matrix?
I'm dealing with a problem that I never met before, in which the question statement is clean & brief, but not sure about the skills/theorem that I should rely on, may be LLN, CLT; or maybe other ...
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