"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)).

learn more… | top users | synonyms

1
vote
1answer
19 views

Error propagation - nonnormal (again)

I have a dataset of ~2000 points. Each of those points has a standard error value associated with it, and it is assumed that the data points and errors are uncorrelated. Both the dataset and the ...
1
vote
0answers
15 views

Central-limit theorem via sample size or sampling magnitude?

I have a small application of the central limit theorem: "Take a sample of 40 exponentially distributed random numbers and calculate their mean. Repeat 1000 times, record each measured mean and ...
0
votes
0answers
3 views

Unclear of details of limit for a Taylor expansion of moment generating function - test this Wed! [migrated]

I keep coming across these limits - the context is moment generating functions and the Central Limit Theorem, but I'm guessing it's a more general question - here is one example (from the proof for ...
2
votes
0answers
22 views

T-test for a small sample with unknown distribution [duplicate]

Consider a simple hypothesis test concerning the mean of a single sample. If the sample is normally distributed and the variance is known, the exact distribution of the sample mean is known ...
3
votes
1answer
101 views

Convergence from Gamma to Normal Distribution

I came across this problem: Problem If I have $X_1, X_2, ..., X_n$ $n$ iid random variables which pdf is $$ f_X(x) = \begin{cases} \dfrac{x^{\mu-1} e^{-x}}{\Gamma{(\mu)}} &0<x<\infty, ...
3
votes
2answers
25 views

Definition of random walk as a summation of independent random processes

I have a complete beginner question on random walk. As per this paper ...
2
votes
1answer
43 views

Asymptotic distribution for moments of gaussian distribution

Is there a way to find the asymptotic distribution for the moments of Gaussian distribution? More specifically, say you have $X_1, ..., X_n \sim N(\mu, \sigma^2)$. For a moment $m_{n, k} ...
0
votes
0answers
12 views

Non parametric test after multiple imputation

In advance, I'm not a statistician and I'm a SPSS user (although I also use STATA fairly well and R rarely). I'm having some trouble in making some nonparametric comparisons after Multiple ...
2
votes
1answer
41 views

Convergence of sequence of random variables

$X$ is the number of heads we got after tossing $n$ fair coins. My question is: $P(X< \frac{n}{2}+\sqrt{n})$ if $n \to \infty$? I tried to apply CLT like this: ...
0
votes
0answers
18 views

central limit theorem applications in risk management

Is there an example of the central limit theorem being used in financial markets, preferably in the field of risk management? I am not able to find applications to financial markets for this ...
14
votes
4answers
609 views

Reasons for data to be normally distributed

What are some theorems which might explain (i.e., generatively) why real-world data might be expected to be normally distributed? There are two that I know of: The Central Limit Theorem (of ...
0
votes
0answers
11 views

Reasons for data to be normally distributed [duplicate]

What are some theorems which might explain (i.e. generatively) why real-world data might be expected to be normally distributed? There are two that I know of 1) The Central Limit Theorem (of ...
3
votes
0answers
40 views

What is the limiting distribution of the sample mean?

My question is relatively simple: what is the limiting distribution of the sample mean? But there are some technicalities I want to discuss. context: I was asked this problem in an exam, and I feel ...
4
votes
1answer
63 views

Do these random variables satisfy Lindeberg's condition?

I have the followig sequences: $Pr(X_n=n)=Pr(X_n=-n)=0.5$ $Pr(X_n=2^{n/2})=Pr(X_n=-2^{n/2})=0.5$ I have to show whether they satisfy Lindeberg's condition or not, but this condition is a bit ...
1
vote
0answers
17 views

Upper bound for sampling fraction for CLT to hold

What is the upper bound for the sampling fraction for the Central limit theorem to hold when sampling without replacement? Context The context for my question is that it is regularly argued (see e.g. ...
5
votes
4answers
76 views

Demonstration of central limit theorem

I teach basic (very) statistics to prisoners in a medium/high security prison and would like to demonstrate the Central Limit Theorem. The classroom has no resources beyond a white board. I can only ...
4
votes
2answers
91 views

Does the central limit theorem apply to these probability density functions?

Let's say you have n uniform random variables from 0 to 1. The distribution of the average of these variables approaches normal with increasing n according to the central limit theorem. What if ...
0
votes
0answers
58 views

What is the multivariate analog of the median?

There is a univariate mean: sum the points and divide by the count. There is a multivariate mean analog - the centroid, a point in a multidimensional space. (1). For the median one sorts the list ...
2
votes
1answer
49 views

mean of population with repeated samples of varying size - how to apply CLT?

I have a population with mean $\mu$ and variance $\sigma^2$. I draw a sample with $n_i$ number of i.i.d random observations from population. I compute the mean for this sample as $mean_i$. I then ...
1
vote
0answers
35 views

Expected value non-independent random variables

Let $X$ be a set of costumers, {$x_1, ..., x_N$}, each $x_i \in X$ have a discount $p_i$ in the interval $[0,1]$, it means if $p_i$ is 0.3, $x_i$ will pay only 0.3 of the entire value. I want to know ...
1
vote
1answer
41 views

Variance in central limit theorem

Why is it that $\sqrt{n}(X_{n}-\mu)$ converges in distribution to $N(0,\sigma^{2})$ but $\sqrt{n}(X_{n}-\mu)/\sigma$ converges in distribution to $N(0,1)$?
0
votes
0answers
15 views

Expectation of ratio of functions of Bernoullis: a concentration question

Consider the following $n \times n$ symmetric matrix of i.i.d. Bernoulli random variables, $X_{ij}$. For $i=1,...,n$ and $i<j\le n$. Let $X_{ij} \sim \text{Bernoulli}(p)$ when $i \ne j$, and let ...
0
votes
1answer
41 views

A simple question on CLT in possible connection with Berry-Esseen thm

I am curious about the contents while I read a note on machine learning. It could be obvious. So, please let me know if I am missing some fundamental things. $X_1,X_2,...,X_n$ are from an i.i.d. ...
1
vote
2answers
277 views

Normal approximation for large data set?

I have a dataset that is highly skewed. See image below: When I transform the data I get the following histogram that makes it look normal: This data however is not normal. I get a p-value of ...
3
votes
0answers
55 views

How to find an appropriate sample size given information about the CV?

You are planning to collect a (simple random) sample to estimate the mean of a non-negative random variable. It is known that the population coefficient of variation (CV) is 1.2. Use the central ...
3
votes
1answer
172 views

Central Limit Theorem, what does it really say? [duplicate]

I know that the Central Limit Theorem states that if a random sample of size $n$ is drawn from iid random variables $X_1, \ldots, X_n$, then the variable $$ Z = \frac{\hat x - E(X) }{{\rm sd}(\hat ...
1
vote
0answers
49 views

ANOVA's and reaction time data

I've been reading a lot on the effect of perceptual load on selective attention but the analysis confuses me somewhat. Some of these studies flag 20-40ms differences between conditions as significant ...
4
votes
2answers
62 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
votes
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
433 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
58 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
49 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
votes
6answers
475 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 ...
7
votes
1answer
210 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
224 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
vote
0answers
46 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
votes
1answer
68 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
vote
1answer
28 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
votes
2answers
74 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
votes
0answers
96 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
140 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
votes
2answers
86 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
93 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
vote
2answers
75 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
51 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 $\sigma^2$ ...
8
votes
3answers
192 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
vote
1answer
60 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
149 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
vote
1answer
90 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
51 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 ...