Questions tagged [central-limit-theorem]

For questions about the central limit theorem, which states: "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)

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Steps in CLT proof unclear

In john rice Mathematical Statistics and Data Analysis we find a proof about the central limit theorem. Let $X_{1} , X_{2}...$ be a sequence of independent random variables having variance $\sigma^{2}$...
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Central Limit Theorem - intuitive explanation without deep math [duplicate]

The Central Limit Theorem says that the distribution of the sample mean is approximately normal. Is there any intuitive explanation for why this should be so? I know it can be proven with deep math, ...
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Where does "Uniform" in "Uniform central limit theorem" come from?

We may all know about the CLT. Today I have seen two articles where the use a new term (to me), that is "Uniform central limit theorem". A uniform central limit theorem and efficiency for ...
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Central limit theorem and strong law of large numbers

I had a question in my mind , if a i.i.d distribution function follows central limit theorem , does that mean it will follow Strong law of large numbers also ?? Since in both cases sample means tends ...
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41 views

$Y$ is equal to the sum of $n$ independent identically distributed Gaussian distribution variables, where $n$ is Poisson distribution

$Y$ is equal to the sum of $n$ independent identically distributed Gaussian distribution variables, where $n$ is Poisson distribution. If $Y$ is approximated to Gaussian distribution, what is its ...
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Derivation of moment generating function for limiting distribution of sum of logbeta distributed variables

A sum of logbeta distributed variables occurs in this question Distribution with a given moment generating function Let, $X_j \sim Beta(j\sigma, 1-\sigma)$, $Y_j = -\log(X_j)$ and $S_n = \sum_{j=1}^n ...
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A statistical model of air pollution

I am currently studying a course in statistical mathemathics at for a degree in engineering. We have this project regarding a fictional scenario of air pollution. The current assignment is to answer ...
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323 views

Why is the z statistic of a binomial proportion test normally distributed?

Suppose there is single sample $X_1\dots X_n$ of binary variable $X\sim Bern(p_1)$ and the following hypothesis is being tested: $$H_0: p_1 = p $$ $$H_1: p_1 \not= p $$ Let us use z-statistic without ...
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Relation between variance, square difference and CLT

NEW EDIT TO CLARIFY THE QUESTION My initial question was about why square difference was used instead of absolute value in the formula of the variance... But I ...
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219 views

Sum of $n$ Poisson random variables with parameter $\frac 1 n$

I have been working through the exercises of a textbook and stumbled upon the question as follows: A skeptic gives the following argument to show that there must be a flaw in the central limit theorem:...
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CLT may fail under this condition?

CLT says that when sample size n goes to infinity, the sample sum or average curve converges to normal distribution. My question is: if the sample size goes closer or even equal to the population size,...
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Convergence in distribution for difference in sample means?

Suppose $X_i, i=1,\ldots, n$ are $i.i.d.$ random variables with mean $\mu_X$ and variance $\sigma^2_X$ $Y_j, j=1,\ldots, m$ are $i.i.d.$ random variables with mean $\mu_Y$ and variance $\sigma^2_Y$ $\...
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Why do we apply the sample mean version of the CLT for a problem involving a sample size of 1?

I am having problems understanding the following question and answer. It seems to me that the sample size is n = 1 and the population size is N=500. If I read it this way then we do not have a large ...
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How to apply Lyapunov CLT to data

I have a situation where I have around 30 classes of variables with different means and variances (though the means aren't too far from eachother; think 4-7) and that the distributions are right ...
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Issues with sampling distribution over bootstrapped monte carlo simulations

Facebook posed an interview question (see ~49 min mark), how many days would it take (in days) to sample every user from a population of 1000, given that you sample 10 users/day each day? Analytically,...
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Intuition behind $2\Phi(x)-1$

When using the central limit theorem to calculate for $n$ I come across having to use $2\Phi(x)-1$ to find this. However, I'm unsure why I have to use this and what it means and how it's derived. It ...
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Testing the Central Limit Theorem with the Shapiro-Wilk test on dice rolling simulations

Rolling a single dice repeatedly will result in a uniform distribution. But if we roll multiple dice the sum would be the Normal distribution due the Central Limit Theorem (CLT). To verify this, ...
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The one-sample t-Test/z-test and inference validity

The CLT tells us that as we collect the means of different samples, the sampling distribution resembles a normal distribution and this way we can infer with a CI on the sampling distribution, the ...
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CLT application to exponential distribution? [duplicate]

I'm a little confused how the CLT can apply to aggregations of the exponential distribution. It's my understanding that the CLT says, in plain English, "sample means from virtually any ...
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279 views

Central Limit Theorem with Bounded Sum of Variances?

I have a sequence of bounded independent random variables $X_1,...,X_n,...$ satisfying $\sum_{i=1}^{\infty} \mathbb{E}[X_i] < \infty$, $\sum_{i=1}^{\infty} Var[X_i] < \infty$. Most versions of ...
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1answer
51 views

Can I use the delta method with a function that depends on n to approximate the distribution of a function of the sum of iid random variables?

Let $X_1, X_2,...$ be i.i.d. random variables with finite mean $\mu$ and finite variance $\sigma^2$. From the Central Limit Theorem, we know that $\sqrt{n}(\bar{X_n}-\mu)$ tends in distribution to $N(...
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59 views

Asymptotic covariance matrix of $\bar{\pmb x}$

In a text I'm reading it says that we define $$ \begin{align} \bar{\pmb x}= \begin{bmatrix}\bar x_1 \\ \bar x_2 \end{bmatrix} \end{align} $$ And then immediately says the asymptotic covariance matrix ...
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59 views

Two Sample Z-test with Non - normal distributions (A/B testing)

I am doing my first steps in statistics and your help will be much appreciated. Sorry for not providing any data, since this is just a made-up exercise that I could not quite find the answer for. Let'...
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192 views

How is it that CLT can be used for around 8 light bulbs?

I have been given the following problem. The light bulbs available have an average lifetime of 1000 hours with a standard deviation of 50 hours. How many light bulbs should we stock so that we can be ...
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Multivariate Lindeberg-Levy CLT, why assuming $E(||X||^2)<\infty$ instead of $E(XX')<\infty$?

Multivariate Lindeberg-Levy CLT(demeaned version) states that "Let $\{X_1,...,X_N\}$ be a random sequence with mean zero, if $E(||X_i||^2)<\infty$, then $\frac{1}{\sqrt{N}}\sum_{i=1}^NX_i$ ...
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71 views

What does "Expectation with respect to true unknown parameter" mean?

I am trying to study the asymptotic properties of MLE, but I am having trouble understanding an expression that seems to be consistently used in all lecture notes available online (page 93,page 18,...
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ISI PCB NC$9$ Limiting Distribution of Bernoulli to Poisson

Let $X_i\sim (i.i.d.)$, Bernoulli($\frac{\lambda}{n}$), $n\ge \lambda\ge 0$. $Y_i\sim (i.i.d.)$, Poisson($\frac{\lambda}{n}$). $\{X_i\}$ and $\{Y_i\}$ are independent. Define $T_n=\sum_{i=1}^{n^2}X_i$...
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CLT theorem and Berry–Esseen bounds for this special case of sampling

Consider a finite set $S=\{s_1,s_2,..s_n\}$, where $a \leq s_i\leq b$ are integers. Each element in $S$ can be chosen to a subset $S'$ in probability $p$. We consider $n$ to be very large. My question:...
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41 views

Distribution of sample means without replacement for large samples

I am starting with $N$ values drawn from a standard Normal distribution. I then sample, without replacement, every subset of size $k$, of which there are $\binom{N}{k}$, and calculate the sample mean ...
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limit behaviour of a quadratic form

I'm reading a book on portfolio optimization and risk management, and I wanna clarify what the author wants to say. Let $\mathbf{X}=[X_1,...,X_n]$ be a random vector with mean $\mathbf{\mu}=E\{\mathbf{...
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141 views

Mean Accuracy and Standard Error of the Accuracy for KNN Classification algorithm

The given below code snippet is from the assignment of online course IBM ML with Python. Here's the assignment. The used variable names :mean_acc and ...
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32 views

Central Limit Theorem: Is the likelihood of obtaining some sample mean exact when n is not infinity?

The central limit theorem states: $$ \lim _{n\to \infty}{\sqrt{n}}{\left({\frac {{\bar {X}}_{n}-\mu }{\sigma }}\right)} \sim \mathcal{N}(0,1) $$ Which means if I ran an infinite number of experiments, ...
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1answer
70 views

If $X_1, ..., X_n$ come from $\exp(5)$ using CLT, Calculate the probability $P(Z<z)$ [closed]

Please advise if the approach i am taking below is correct, I don't see how else to go about solving this problem. ...
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1answer
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What, if any, asymptotic arguments are used in moving between various statements of the central limit theorem?

What, if any, asymptotic arguments are used in moving between the various statements of the central limit theorem (e.g. in terms of sample means compared with standardised sample means)? Context. My ...
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1answer
132 views

Question about the central limit theorem? sample size

I am using the following definition of the central limit theorem: Suppose $X_1 ,X_2\dots,X_n$ are indepdent identical with $E(X_i)=\mu$ and $Var(X_i)=\sigma^2$. Then as $n\to\infty$, $Z_n=\frac{X_1+...
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1answer
81 views

How to derive the form of the central limit theorem for the difference-in-means estimator?

In randomized studies, we have that the difference-in-means estimator is given, for treatment/control as: $$ \hat{\tau}_{DM} = \frac{1}{n_1}\sum_{Z_i = 1}Y_i - \frac{1}{n_0}\sum_{Z_i = 0}Y_i $$ where $...
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How to know the number of subjects and events required for statistical significance? [duplicate]

Say I conduct a study where i measure physiological signal of a set of N subjects, to record a particular event. I am interested of the duration and the amplitude of that event. The study ends when I ...
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164 views

How can the CLT fix OLS regression residuals that are not normally distributed?

I often hear that when the residuals depart from normality, the central limit theorem can be used to fix things. I do not quite understand how this works, since the central limit theorem is a ...
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Central limit theorem in practice and Berry-Essen

In a first course to statistics, confidence interval calculation using the Central Limit Theorem is introduced. Are confidence intervals sensible in practice? That is, given a 99% confidence interval ...
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298 views

why n>=30 for central limit theorem to hold? [duplicate]

From population choosing samples(size n=30) and calculate its mean then repeating it N times will converge to normal distribution as N->inf when mean of each sample is plotted as a histogram. From ...
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2answers
148 views

How to understand taking a simple sample to compute the Confidence Interval, using CLT?

Suppose there is a population distribution having mean M and standard deviation SD. We want to estimate the Confidence Interval by using a single sample using CLT. We take a sample and compute it's ...
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1answer
32 views

Non-normal sample from a non-normal population (option returns) does the central limit theorem hold?

I'm testing a short call option strategy and found, as expected, non-normal return distributions. It is known that option returns are not normally distributed (i.e., also the population). I take the ...
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1answer
42 views

Proof Poisson converges to Normal [closed]

I am looking for a formal proof that, with the CLT transformation, a random variable $Y \sim POI(\lambda)$ converges to a normal distribution ($Z\sim N(0,1)$). I believe this can be formulated as: $$...
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Original vs Sample mean distribution in reconstructing time series data

Lets assume that the time series corresponds to some observable signal from a processor that computes a modulo 2 operation. The nature of the time series depends largely on the output of the modulo ...
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What is the meaning behind a sample distribution if you only have one sample?

I'm trying to understand the meaning behind the central limit theory and the importance of CLT for inferential statistics. The problem that I encountered has to do with sample distributions. ​ I do ...
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2answers
28 views

Sampling as adding random variables, especially binomial RVs

Is sampling equivalent to adding random variables? I'm a bit confused because as we can see that the binomial distribution becomes more and more shaped like a normal distribution as $n$ increases. We'...
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29 views

PDF of given complex exponential equation of random variables [closed]

Let $d_i$ and $d_k$ both are independent and identical random varibles with pdf $\frac{3d_i^2}{R_{max}^3-R_{min}^3}$ where $R_{max}$ and $R_{min}$ both are constants and $ R_{min}<d_i<=R_{max}$...
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CLT in Diebold & Mariano (1995)

The Diebold-Mariano (DM) statistic is derived as follows: Assuming the loss-differential between the two models $d_t$ is covariance stationary: $$\begin{cases} \mathbb{E}[d_t] = \mu> 0 & \...
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1answer
72 views

CLT so we have enough burgers!

An event has 32 people. Caterers need to make burgers for these people. They expect that a person at this event might need 0, 1 or 2 burgers with probabilities 0.2, 0.55, and 0.25 respectively. We ...
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1answer
29 views

Picking a specific estimated CDF from a set of CDFs provided by an ECDF

Let $F_X$ be a CDF of an unknown random variable $X$. If we have independent samples $x_1, x_2, \ldots, x_n$ of $X$ then we can estimate $F_X$ non-parametrically using an ECDF $\hat{F}_n$. By Central ...

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