The central-limit-theorem tag has no wiki summary.
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Confidence intervals based on the CLT: ever useful?
Suppose, for concreteness, that I am trying to estimate the mean of a population using a random sample of size $N$.
Many elementary books discuss forming a confidence interval for the population mean ...
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1answer
50 views
Use the Central Limit Theorem to calculate the approx probabilities of Gamma RVs?
If $X_i, i=1,...,$ are independent, identically distributed $\operatorname{N}(0,1)$ random variables, and $Y_i = X_i^2$ are independent $\operatorname{Gamma}(\frac{1}{2},\frac{1}{2})$ RVs, use the ...
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3answers
217 views
What's the distribution of $\bar{X}^{-1}$?
What's the distribution of $\bar{X}^{-1}$ with X being a continuous iid random variable that is uniformly distributed? Can I use the CLT here?
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Can the proof of the central limit theorem be expressed as the limit of a convolution with an operator
The central limit theorem, as I understand it (engineer, non-statistician), says that the distribution comprised of means of some other reasonably behaved distributions converges to a normal ...
8
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1answer
288 views
Central limit theorem and the law of large numbers
I have a very beginner's question regarding the Central Limit Theorem (CLT):
I am aware that the CLT states that a mean of i.i.d. random variables is approximately normal distributed (for $n \to ...
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1answer
71 views
Berry-Esseen theorem
In the Berry-Essen theorem, why is the standard normal distribution used in the context of the closedness of two distributions? Why can't we use the general normal distribution where will things will ...
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1answer
133 views
Evaluate $\lim_{n \to \infty} \sum_{j=0}^{n}{{j+n-1} \choose j}\frac{1}{2^{j+n}}$
Using the Central Limit Theorem , Evaluate $$\lim_{n \to \infty} \sum_{j=0}^{n}{{j+n-1} \choose j}\frac{1}{2^{j+n}}$$
My solution: Let $\{X_n\}$ be a sequence of iid R.V's each having ...
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1answer
78 views
Find the limiting distribution of $Z_n$
Let $X_1,X_2, \dots$ be iid RVs with mean $\alpha$ and variance $\sigma^2$ , and let $Y_1,Y_2, \dots$ be iid RVs with mean $\beta(\neq 0)$ and variance $\tau^2$. Find the limiting distribution of ...
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1answer
289 views
Central limit theorem for sample medians
If I calculate the median of a sufficiently large number of observations drawn from the same distribution, does the central limit theorem state that the distribution of medians will approximate a ...
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0answers
72 views
Can we approximate the distribution of S?
I want to understand how the sampling distribution of the whole covariance matrix behaves for large $n$. I am trying to use the delta method and multivariate CLT. I am trying to show that when the ...
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1answer
293 views
When to use Central Limit Theorem?
Is the central-limit-theorem true for all distributions?
For what sample size is it true (what is meant with "large")?
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Empirical coverage probabilities of confidence intervals
Can someone tell me how the type of distribution is related to an empirical coverage probability? (for some distributions the Central-limit-theorem seems to be more easily satisfied)
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130 views
Central limit theorem with unknown variance
In my experiment I compute the average Latency of operations per second.
Now, I would like to define N, i.e: how many times do I need to run my experiment to compute a close to real average latency.
...
8
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1answer
255 views
Does the multivariate Central Limit Theorem (CLT) hold when variables exhibit perfect contemporaneous dependence?
The title sums up my question, but for clarity consider the following simple example. Let $X_i \overset{iid}{\backsim} \mathcal{N}(0, 1)$, $i = 1, ..., n$. Define:
\begin{equation}
S_n = \frac{1}{n} ...
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0answers
37 views
Central Limit Theorem [duplicate]
Possible Duplicate:
What intuitive explanation is there for the central limit theorem?
I am in an introductory statistics course, and I am having trouble understanding the Central Limit ...
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1answer
54 views
A limit theorem for non-independent variables
Let $X_n$ be a sequence of identically distributed (e.g., binomial $B(1,1/2)$) random variables which are not independent (say, for any $n$ and $m$, $corr(X_n,X_m)=c$).
What can be said about the ...
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Is it OK to use the CLT to create a normal distribution where there is none?
I have some data that looks like this:
Procrastinator has come up with one good suggestion for how to test hypotheses under this distribution, but it relies on some guesswork to fit constants. I ...
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3answers
791 views
Consider the sum of $n$ uniform distributions on $[0,1]$, or $Z_n$. Why does the cusp in the PDF of $Z_n$ disappear for $n \geq 3$?
I've been wondering about this one for a while; I find it a little weird how abruptly it happens. Basically, why do we need just three uniforms for $Z_n$ to smooth out like it does? And why does the ...
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1answer
87 views
Weak convergence of an empirical process. Demonstration
The empirical process $B_n(x) = \sqrt{n}(F_n(x) − F(x))$ converges weakly to a zero-mean Gaussian process, $B$, with covariance function:
$\mbox{cov}(B(x), B(y)) = F(\min\{x, y\}) − F(x)F(y)$.
How I ...
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4answers
454 views
Where does $\sqrt{n}$ come from in central limit theorem (CLT)?
A very simple version of central limited theorem as below
$$
\sqrt{n}\bigg(\bigg(\frac{1}{n}\sum_{i=1}^n X_i\bigg) - \mu\bigg)\ \xrightarrow{d}\ \mathcal{N}(0,\;\sigma^2)
$$
which is Lindeberg–Lévy ...
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Cental limit theorem for average of iid variables
The Wikipedia entry on the CLT states at one point: "For fixed large $n$ one can also say that the distribution of $S_n$ is close to the normal distribution with mean $\mu$ and variance ...
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252 views
“Central limit theorem” for weighted sum of correlated random variables
I'm reading a paper which claims that
$$\hat{X}_k=\frac{1}{\sqrt{N}}\sum_{j=0}^{N-1}X_je^{-i2\pi kj/N},$$
(i.e. the Discrete Fourier Transform, DFT) by the C.L.T. tends to a (complex) gaussian random ...
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4answers
1k views
Which test to choose when the results from t-test and Wilcoxon test are different?
I have a sample of 48. According to the central limit theorem, I may consider means of every continuous variable in my sample to have a normal distribution.
However, one variable has a mean of 14 +/- ...
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3answers
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Information theoretic central limit theorem
The simplest form of the information theoretic CLT is the following:
Let $X_1, X_2,\dots$ be iid with mean $0$ and variance $1$. Let $f_n$ be the density of the normalized sum $\frac{\sum_{i=1}^n ...
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1answer
474 views
Error in normal approximation to a uniform sum distribution
One naive method for approximating a normal distribution is to add together perhaps $100$ IID random variables uniformly distributed on $[0,1]$, then recenter and rescale, relying on the Central Limit ...
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2answers
180 views
How/Why does resampling from “any” distribution lead to a normal distribution?
I was performing some Monte-Carlos on historical data and irrespective of the distribution of the data I would always get a normal distribution owing to resampling with replacement. That made it easy ...
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4answers
268 views
The operation of chance in a deterministic world
In Steven Pinker's book Better Angels of Our Nature, he notes that
Probability is a matter of perspective. Viewed at sufficiently close
range, individual events have determinate causes. Even a ...
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3answers
187 views
CLT and stable distributions
I have a few questions about generalizations of the CLT and stable distributions. I'm trying to correct my understanding and make it precise. Please forgive my naivete, I am not a professional ...
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1answer
154 views
Accuracy in approximation with central limit theorem
I have set of random values with the same distribution $y_1, \ldots, y_N$
$$T = \frac{1}{N}\sum_{j = 1}^{N} y_j$$
I want to find the confidence interval for the mathematical expectation $E(Y)$
...
4
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1answer
181 views
Proofs of the central limit theroem
I know there are different versions of the central limit theorem and consequently there are different proofs of it. The one I am most familiar with is in the context of a sequence of identically ...
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265 views
Whether to use nonparametric tests to compare two groups when sample size is large but assumptions are violated
I have a data set that has 600 observations divided in two groups. I am going to compare the central tendencies (e.g., the means) of these two groups. However, there are violations of classical ...
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Why law of large numbers does not apply in the case of Apple share price?
Here is the article in NY times called "Apple confronts the law of large numbers". It tries to explain Apple share price rise using law of large numbers. What statistical (or mathematical) errors does ...
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3answers
2k views
Central limit theorem versus law of large numbers
The central limit theorem states that the mean of i.i.d. variables, as $N$ goes to infinity, becomes normally distributed.
This raises two questions:
Can we deduce from this the law of large ...
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1answer
445 views
Understanding central limit theorem
What is wrong with the following sentence:
The Central Limit Theorem implies that, as the sample size grows, the error distribution approaches normality.
Am I correct by saying that it should ...
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2answers
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Assess density of a function in non-closed form
I am working with Rice's Mathematical Statistics and Data Analysis and on page 179 it introduces Monte Carlo integration. This question is a smallish revision based on comments from @Iterator (in R ...
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How many of the biggest terms in $\sum_{i=1}^N |X_i|$ add up to half the total?
Consider $\sum_{i=1}^N |X_i|$
where $X_1, \ldots, X_N$ are i.i.d. and the CLT holds.
How many of the biggest terms add up to half the total sum ?
For example, 10 + 9 + 8 $\approx$ (10 + 9 + 8 $\dots$ ...
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409 views
How to demonstrate failure of CLT in R?
I was assigned to demonstrate the Central Limit Theorem (CLT) in R in my statistics class. I already made some progress with simulation using simple.sim in R. I want to prepare 3 examples of the CLT ...
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How to test for differences between two group means when the data is not normally distributed?
I'll eliminate all the biological details and experiments and quote just the problem at hand and what I have done statistically. I would like to know if its right, and if not, how to proceed. If the ...
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2answers
333 views
Putting a confidence interval on the mean of a very rare event
I'm simulating an extremely rare event (the detection of weighted photon packets in a highly absorbing material). For example, I may simulate the transmission of 1e9 photons and only detect 10 of them ...
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1answer
642 views
Central limit theorem for sample quantiles
I am reading Ruppert's Statistics and Data Analysis for Financial Engineering, which contains the following theorem:
Let $Y_1$, $...$ , $Y_n$ be an i.i.d. sample with a CDF $F$. Suppose
that ...
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1answer
373 views
Random walks in multinomial case
Model:
A vector $X=(X_1, X_2, X_3)$ that follows a trinomial distribution with parameters $p=1/3$ and $n$.
(I have a coin with three sides $S1$, $S2$, $S3$). I flip the coin $n$ times. The coin has ...
4
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1answer
135 views
Does the Central Limit Theorem allow one to create confidence intervals from a web traffic dataset?
I'm a journalist turned developer who hobbies in APIs and analysis of web traffic. I've always enjoyed learning about stats but as I learned, I learned that I have misapplied some basic concepts in ...
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1answer
365 views
Conditions for Central Limit Theorem for dependent sequences
Cumbersome technical assumptions (e.g., mixing properties) are used in the literature to prove Central Limit Theorems for dependent sequences. I sketched a proof that does not require any of these ...
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Summing normal instead of beta distributions, consequences for the density function of the sum?
Background:
I've modeled a project effort prediction as a Google Spreadsheet template. Details of the Model: http://sites.google.com/site/effortprediction/methodology
.Google Spreadsheet does not ...
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2answers
283 views
Central limit theorem for sum from varied distributions
The central limit theorem as I am familiar with it applies to the limiting (rescaled) distribution of $n$ convolutions of a single probability distribution as $n$ goes to infinity, or equivalently, to ...
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1answer
163 views
Estimate the nearest of N random points in a box in E^d?
I have N uniform-random points $p_j$ in a box in $E^d$,
$a_i \le x_i \le b_i$,
and want to estimate the expected distance of the point nearest the origin in $L_q$:
$\quad$ nearest( points $p_j$, box ...
6
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1answer
230 views
Limiting distribution of a squared sum of random variables
Let $S_n = \frac{1}{n}\sum_{i=1}^n X_i$, and $T_n = \frac{1}{n}\sum_{j=1}^nY_i$, where
The $X_i$ are iid, the $Y_i$ are iid (with a different law)
$X_i$, and $Y_i$ are dependent
For $i\neq j$, ...
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3answers
630 views
What is the sum of squared t variates?
Let $t_i$ be drawn i.i.d from a Student t distribution with $n$ degrees of freedom, for moderately sized $n$ (say less than 100). Define
$$T = \sum_{1\le i \le k} t_i^2$$
Is $T$ distributed nearly as ...
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1answer
248 views
Central Limit Theorem Tails
I'm trying to compute some p-values for samples from a distribution of sums of ~1000 random variables. The exact distribution of these random variables isn't known, but I have empirical estimates that ...
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382 views
How to know which probability distribution to expect?
Currently I am analyzing social networks (not virtual) and I am observing the connections between people. If a person would choose another person to connect with randomly, the number of connections ...
