# Tagged Questions

"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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### why does this election vote converge to bell curve?

Law of large numbers central limit theorem says IID random variables deviate from their average like a Gaussian, but these state matchups have different outcomes and probabilities. In fact, they are ...
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### How can we get a normal distribution as $n \to \infty$ if the range of values of our random variable is bounded?

Let's say we have a random variable with a range of values bounded by $a$ and $b$, where $a$ is the minimum value and $b$ the maximum value. I was told that as $n \to \infty$, where $n$ is our sample ...
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### Drawing sample size 1 from a uniform distribution - Difficulty in understanding Central Limit Theorem?

Let's say we have a uniform distribution and we are drawing samples of size 1 so that the mean is the drawn number itself. If we perform this activity sufficiently large number of times we would get ...
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### Why are all CLT problems using a single random sample when the CLT requires “repeatedly drawing random samples”?

In our course notes, it says: "The CLT states that if random samples of size $n$ are repeatedly drawn from any population with mean $\mu$ and variance $\sigma^2$, then when $n$ is large the ...
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### CLT and Wald-Wolfowitz runs test asymptotic distribution

I need help finding a theorem which could be used to prove that the Wald-Wolfowitz runs test is asymptotically normal. Let me formalize my question. We have a random sample $\{X_0,X_1,...,X_n\}$ (if ...
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### When is it appropriate to use the Central Limit Theorem?

I am currently having a read through the Statistical Drake Equation; a method of taking the Drake Equation, letting each number be a uniform random variable, and then applying the Central Limit ...
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### find variance using CLT (Central Limit Theorem)

the question is A liquid drug is marketed in phials containing a nominal 1.5ml but the amounts can vary slightly. The volume in each phial may be modeled by a normal distribution with the mean 1.55ml ...
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### Interpretting Confidence Interval Questions

Q.In a survey conducted by a mail order company a random sample of 200 customers yielded 172 who indicated that they were highly satisfied with the delivery time of their orders. Calculate an ...
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### Role of Central Limit Theorem in creating confidence intervals and hypothesis testing

How is central limit theorem useful in creating confidence intervals as well as its role in hypothesis testing around the population parameter.
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### convergence of geometric mean/harmonic mean

Does any one know papers regarding the convergence of geometric mean or harmonic mean in probability, parallel to central limit theorem?
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### Is the sum of a large number of independent Cauchy random variables Normal?

By Central Limit Theorem, the probability density function of the the sum of a large independent random variables tends to a Normal. Therefore can we say that the sum of a large number of independent ...
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### Sample Size vs Iterations (CLT)

I am slightly confused by the term 'sampling size'. Let's say we have N-dimensional distribution and we take one sample (i.e., one vector of N-elements) and measure it's mean. Now let's do 100 ...
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### Understanding Central Limit Theorom - Sample Deviation

If only the standard deviation of sample is known and not the standard deviation of the population, you don't have to square root the sample size to find the value of Z score for CLT?
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### Why does the central limit theorem work with a single sample?

I have always been taught that the CLT works when you have repeated sampling, with each sample being large enough. For example, imagine I have a country of 1,000,000 citizens. My understanding of CLT ...
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### A dynamical systems view of the Central Limit Theorem?

(Originally posted on MSE.) I have seen many heuristic discussions of the classical central limit theorem speak of the normal distribution (or any of the stable distributions) as an "attractor" in ...
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### Central Limit Theorem when the dimension size increases with the sample size

Let $X_1, X_2,\ldots, X_n \in \mathcal{R}^d$ and be zero-mean, unit variance random variables. Here the dimension ($d$) is a function of the sample size($n$) i.e, $d=f(n)$. For example $d = \sqrt{n}$. ...
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### Understanding formulas for the sampling distribution of the mean

In the passage below, what does $k_c$ mean, and why (in "$σ_0/\sqrt n$") is $\sigma$ being divided by the square root of $n$? I got this form the book Principles of statistical inference.
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### Basic question about central limit theorem and sampling distributions

The CLT states that "if you have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population, then the distribution of the sample means will be ...
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### Normality test in panel data analysis

Is it necessary to perform normality test such as Jarque Bera, Kolmogrove etc in panel data analysis? Assume I have 3000 obervation..Since the observation size is big, can I just assume that the data ...
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### Independent samples t-test: Do data really need to be normally distributed for large sample sizes?

Let's say I want to test if two independent samples have different means. I know the underlying distribution is not normal. If I understand correctly, my test statistic is the mean, and for large ...
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### Distribution of samples from a uniform distribution [duplicate]

Let's say we are taking $n$ samples from a uniform distribution, that spans from $0$ to $1$. According to the central limit theorem, the mean of the $n$ samples will follow a normal distribution with ...
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### Central limit theorem and the Pareto distribution

Can somebody please provide a simple (lay person) explanation of the relationship between Pareto distributions and the Central Limit Theorem (e.g. does it apply? Why/ why not?)? I am trying to ...
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