"Given certain conditions, the mean of a sufficiently large number of iterates of independent random variables, each with a well-defined mean and well-defined variance, will be approximately normally distributed" ([Wikipedia](http://en.wikipedia.org/wiki/Central_limit_theorem)).

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4
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2answers
43 views

Sampling distribution of the mean of population that assume values only between 0 and 1

I'm trying to use central limit theorem to compute the mean of the sample, however the population where I'm sampling has value only between 0 and 1, can I use mean of the sample as mean of the ...
0
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0answers
10 views

Normal distribution approximation using central limit theorem [duplicate]

I have a variable with 150 observations and it is non normal. Wouldn't CLT be useful in this context. What does large n really mean in CLT?
3
votes
1answer
31 views

Confidence interval for the standard deviation on a bimodal distribution

I have a bimodal distribution, and I wish to estimate the mean and standard deviation of the population (well, these could be 2 sub-populations according to the shape). With the mean I have no ...
2
votes
0answers
46 views

Standard error on median for exponential distribution

I am trying to find the standard error on the median, $\sigma_\tilde{x}$, for a sample, $X_i$, of a population whose pdf could be modelled as $\lambda e^{-\lambda x} $ if normalized. To make sure we ...
5
votes
1answer
39 views

Reference request: local limit theorem for log-concave densities

The following is easy to prove and can't possibly be new. But I can't find it printed anywhere despite some effort. Can anyone tell me where it is published? Let $X_1,X_2,\ldots$ be a sequence of ...
6
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6answers
343 views

Dealing with non-normal distribution in “big” datasets, when do we throw out the CLT?

Apologies from the go as this question comes from an absolute newbie and will definitely not satisfy a lot of the detail required. Hence, your guidance in providing you the right information to allow ...
5
votes
1answer
52 views

Asymptotic distribution of sample variance of non-normal sample

This is a more general treatment of the issue posed by this question. After deriving the asymptotic distribution of the sample variance, we can apply the Delta method to arrive at the corresponding ...
3
votes
1answer
144 views

Question about standard deviation and central limit theorem

I have a quick question about the central limit theorem. Lets say I measure some value that comes from an arbitrary distribution N times and I repeat this M times. I understand that if I calculcate ...
1
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0answers
31 views

Probability that a sample mean is between two values using Central Limit Theorem

Mean is $2.707$, standard deviation is $.049$, sample of $35$ is drawn from the population. What is the probability that the mean price for the sample was between $2.683$ and $2.716$? It was suggested ...
0
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1answer
57 views

Quick Question - Approximate distribution for sample mean?

I am having issues answering part two. I think it is about CLT, correct me if I'm wrong. But how do you compute the distribution details from this ? Please help and thank you for all your ...
1
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1answer
23 views

Use of standard error of the mean when the components are not identically-distributed

Let {$X_1$, ..., $X_n$} be a random sample of size n in which the different elements are measurements drawn from $m$ different populations with different distributions. From the Central Limit Theorem, ...
0
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2answers
60 views

How do I test that a sequence of data satisfies the Central limit theorem?

I have some data stored as a list from some computations. I want to calculate the "distance" the sequence is away from satisfying the central limit theorem. I would like to know, what should I use as ...
0
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0answers
37 views

Error for combining multiple binomial distributions

This problem is somewhat involved and I have a partial solution so bear with me. I will illustrate the problem with an example. Lets say we have two processes and we want to know which has a higher ...
7
votes
1answer
113 views

Is there a theorem that says that $\sqrt{n}\frac{\bar{X} - \mu}{S}$ converges in distribution to a normal as $n$ goes to infinity?

Let $X$ be any distribution with defined mean, $\mu$, and standard deviation, $\sigma$. The central limit theorem says that $$ \sqrt{n}\frac{\bar{X} - \mu}{\sigma} $$ converges in distribution to a ...
0
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2answers
47 views

Self-study question on CLT

Good evening everyone, I am currently doing a self-study on CLT and was working on an exercise which asks if the following statement is true or false CLT guarantees that the population mean is ...
2
votes
2answers
58 views

Convergence in distribution results not deriving from central limit theorem?

In the stats class I took, all the results I have encountered about the convergence in distribution of some random variables are in one way or another consequences of the Central Limit Theorem. Out ...
1
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2answers
58 views

Normal Distribution CLT Question

I am working on a self-study question where A study indicates that the typical American woman spends USD 340 per year for personal care products. The distribution of the amount follows a ...
0
votes
1answer
39 views

Distribution of a Mean and Variance

Say we have observations $x_1 \dots x_n$ and we have some sort of Bayesian framework where we would like to estimate a distribution for the mean $\mu$ of our observations and the variance $\sigma^2$ ...
7
votes
3answers
167 views

If $Z_i =\min \{k_i, X_i\}$, $X_i \sim U[a_i, b_i]$, what is the distribution of $\sum_iZ_i$?

Assume the following set up: Let $Z_i = \min\{k_i, X_i\}, i=1,...,n$. Also $X_i \sim U[a_i, b_i], \; a_i, b_i >0$. Moreover $k_i = ca_i + (1-c)b_i,\;\; 0<c<1$ i.e. $k_i$ is a convex ...
1
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1answer
48 views

Theoretical Justification for Cross Validation

I get it, cross validation works. I'm wondering if there is an literature out there giving any theoretical justification for cross validation. My thought is that there should be, at least, something ...
5
votes
1answer
127 views

Using central limit theorem for approximation

Let $X$ be a random varaible from a distribution with pdf $$ f(x) = \theta x^{\theta-1}, \quad 0< x < 1. $$ a) Name the distribution of $U=-\ln(X)$ by first finding its density ...
1
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1answer
76 views

Does heteroskedasticity matter if you have a large enough sample?

Let's say you run a regression with over 200 observations. Would this reasonably large sample mitigate the impact of residuals heteroskedasticity as an offshoot of the Central Limit Theorem, or ...
-1
votes
1answer
35 views

Can I pull a confidence interval out of a single sample by dividing it into sub-samples?

Let's say I have taken a sample $S$ of a population. I am trying to figure out the population mean. Because I have only made one sample, the best I can do is assume that $\bar S$ is the mean of the ...
1
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1answer
59 views

Does the Central Limit Theorem only work for iid random variables?

Can we say anything about the distribution of the sum of not iid random variables?
7
votes
2answers
197 views

Convergence in Distribution\CLT

Given that $N = n$, the conditional distr. of $Y$ is $\chi ^2(2n)$. $N$ has marginal distr. of Poisson($\theta$), $\theta$ is a positive constant. Show that, as $\theta \rightarrow \infty$, $\space ...
5
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0answers
74 views

Example of CLT when moments do not exist

Consider $X_n = \begin{cases} 1 & w.p (1 - 2^{-n})/2\\ -1 &w.p~ (1 - 2^{-n})/2\\ 2^{k} &w.p~ 2^{-k} \text{ for } k > n\\ \end{cases}$ I need to show that even though this has ...
2
votes
1answer
65 views

CLT can be used for weighted sum of different Bernoulli variables?

Suppose $$ z_i \sim Bernoulli (p_i) $$ Can we use CLT for the following weighted sum? $$ S = \sum_i w_i z_i $$ i.e. can $S$ be approximated with a normal distribution? If yes, with which theorem? ...
1
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1answer
96 views

central limit theorem

can I do my statistics work based on the central limit theorem? I need to perform a t-test, ANOVA and multiple regression. my outcome variable is highly not normally distributed (Highly positively ...
-3
votes
1answer
60 views

Self-study question

I'm currently working on a self study worksheet. I understand most parts of the solution for part III, but I can't seem to make out how this comes about: QUESTION: ANSWER:
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0answers
40 views

Central limit theorem with scilab

I'm trying to illustrate the CLT with scilab but my results are weird. Did I make a mistake ? ...
3
votes
0answers
49 views

Shouldn't a function of data from a PDF repeated over and over on new data eventually yield a Gaussian PDF?

I got into an interesting discussion with a co-worker today and we are not sure what the answer is: We have $N=1000$ samples from a Rayleigh PDF. We take those $N$ samples, and compute their ...
2
votes
1answer
52 views

Central Limit Theorem: Likelihood multiplication

Been watching this video by Tom Minka on Expectation Propagation (http://videolectures.net/mlss09uk_minka_ai/). At about 19:12, he says that the reason the moment matching technique works is that when ...
3
votes
3answers
152 views

Can a normal approximation assumption justify itself?

I am learning elementary statistics. I found an exercise, which asks to compute the desired sample size for some interval for standard error. The solution, in class slides, first assumes the sample ...
2
votes
0answers
104 views

Convergence in distribution (central limit theorem)

If $X_1, ... , X_n$ be iid exponential with mean $1/\lambda$. Let $S_n = X_1 + ... + X_n$. a) Show that $S_n$ is $\Gamma(n, 1/\lambda)$. Each $X_i$ is $\Gamma(1, 1/\lambda)$ by the ...
1
vote
1answer
136 views

Can a two-sample t-test be used with data that doesn't follow a normal distribution?

One of the assumptions for t-tests is that the data must follow a normal distribution. However, due to the Central Limit Theorem (and this thread): "if the sample is large enough you can use t-test ...
4
votes
2answers
192 views

How useful is the CLT in applications?

My lecturer just covered the Lindberg-Levy central limit theorem and the multivariate version, the Lindberg-fuller CLT. I understood the basic concept and I can derive it, etc. But it would help my ...
0
votes
1answer
69 views

What does it mean to scale random variables?

We just started learning asymptotic theory, and to prove the lindberg-levy central limit theorem, weak law of large numbers etc, we 'scale and standardize' the RVs so it ends up having a standard ...
0
votes
0answers
30 views

Probability of an SRS having a mean between two values

Can anybody help me out with the following question? The distribution of the age at which all married males got married is right skewed with mean = 22.9 years and  standard deviation = 1.5 years. ...
0
votes
1answer
78 views

When Monte Carlo simulation can't be used to simulate a statistical system?

My question is simple. Which are the general conditions for which a Monte Carlo simulation can be used to represent a statistical system? Or conversely, which are the statistical system that cannot be ...
3
votes
1answer
81 views

Law of Large numbers and central limit theorem

I have come across a bit of a dilemma in my exam revision, this is not a topic I am particularly strong with so help is most appreciated: Assume that $X_1,X_2,...$ is an i.i.d sequence, where ...
1
vote
1answer
92 views

Does the central limit theorem imply the law of large numbers

Assuming that the distribution has finite variance (a condition not required for the LLN), then doesn't the LLN follow from the CLT?
1
vote
1answer
85 views

What characteristics of the distribution of a test statistic can be inferred using a bootstrap?

Setup: Let $p_n(x) = \mathbb{P}(S_n \leq x)$ and let $p_n^*(x) = \mathbb{P}^*(S_n^* - S_n \leq x)$, where $S_n$ is some zero-mean test statistic that can validly be bootstrapped, eg a sample mean, and ...
5
votes
2answers
286 views

When combining p values, why not just averaging?

I recently learned about Fisher's method to combine p-values. This is based on the fact that p-value under the null follows a uniform distribution, and that $$-2\sum_{i=1}^n{\log X_i} \sim \chi^2(2n), ...
2
votes
0answers
54 views

Why does the accuracy of the central limit theorem for a mean depends on the skewness of the random variables being summed?

The accuracy of the central limit theorem for a mean depends on the skewness of the random variables being summed. Could someone explain that to me? Or could someone propose a book to read? ...
1
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0answers
73 views

Central Limit Theorem - interval estimation

I'm rolling a regular dodecahedron (12-side die) 1200 times. I need to find an interval, in which the total count of prime-number results will lie with the probability of 95%. I have to use the ...
0
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0answers
63 views

CLT in a Monte Carlo simulation, small sample

A CLT says that asymptotically the sampling distribution of the sampling mean converges to the Normal. I would like to run a Monte Carlo simulation using information on one of the model's variables ...
1
vote
3answers
126 views

Distribution of a sample from a normal population

If I were to collect ONE sample of size 20 from a normal population, how justified would I be in claiming that the sample is normally distributed? I'm getting a little confused since by the Central ...
1
vote
1answer
88 views

Rate of convergence for SLLN

I am interested in writing a non-asymptotic rate of convergence for SLLN as a function of number of samples. From the literature I've read so far, CLT provides an asymptotic convergence rate of ...
4
votes
0answers
75 views

Is there a statement for the median like the central limit theorem for the mean? [duplicate]

According to the Central Limit Theorem (CLT) in statistics, the distribution of the average of randomly sampled n observations tends to follow normal distribution as the sampling size n becomes ...
3
votes
2answers
132 views

Taking the log of variables

Just before I start the question I would like you all to know that I have checked the other threads on taking the log of variables but I still think I have a question that hasn't been touched on yet. ...