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Questions tagged [central-limit-theorem]

This tag should be used for each question where the term "central limit theorem" and with the tag (tag:probability-limit-theorems). The central limit theorem states that the sampling distribution of the mean of any independent, random variable will be normal or nearly normal, if the sample size is large enough.

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104 views

Show $X$~Exponential distribution given $P(X > nx) = (P( X > x ))^n$

Show $X$~Exponential distribution given $P(X > nx) = (P( X > x ))^n$ Can someone give some hint ? The only thing I can think of is to write it as CDF format and say according to observation ...
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When do we use the Central limit theorem to construct a confidence interval? [closed]

When should I use the Central limit theorem to construct a confidence interval? What kind of parameter would I be "estimating" with the CLT confidence interval? Thanks for any help!
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Sum of sequence of non-identical random variables. Can they become something else than Normal distributed?

The Central Limit Theorem is well known in statistics. It states: ...
30 views

Convergence for ${\bf p}^H {\bf D}{\bf p}$ using law of large numbers

Suppose ${\bf p} = [p_1, \dots, p_n]^T$ be a $\mathbb{C}^{n \times 1}$ vector whose elements are i.i.d zero-mean and unit variance random variable (RVs), i.e., $\mathbb{E}[|p_i|^2] = 1$. Then from law ...
56 views

Convergence for $\frac{1}{n} {\bf p}^H {\bf D}{\bf q}$ using law of large numbers

Suppose ${\bf p} = [p_1, \dots, p_n]^T$ and ${\bf q} = [q_1, \dots, q_n]^T$ be mutually independent $\mathbb{C}^{n \times 1}$ vectors whose elements are i.i.d zero-mean and unit variance random ...
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Normal distribution and sample distribution question

In a recent year, the distribution of scores of students on the ACT college entrance exam was modelled by a normal distribution with a mean of 20.9 and a standard deviation of 4.7. 1) The mean score ...
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Controlling the Lyapunov condition

I have been struggling with the following exercise for quite some time now. Let $(Z_n)_{n \geq 1}$ be a sequence of independent random variables such that for $j=1,2, \ldots$ we have that for some ...
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Equidistributed sequences satisfying the central limit theorem

For a sequence (x_n) of reals that are equidistributed modulo one (such as (prime-) multiples of irrationals, or most geometric sequences) does the (properly scaled) sum of the remainders converge to ...
97 views

Central limit theorem for weighted average

Let $(a_i)_{i\ge1}$ be a bounded positive sequence and $X_i$ be iid random variables with mean $0$ and finite variance. Let $s_n=\frac{\sum_{i=1}^n a_i X_i}{\sqrt{\sum_{i=1}^n a_i}}$. If $a_i=1$ for ...
37 views

Central Limit theorem with extremely skewed population

If $X$~$N(1,0.1)$, and $Y=X^n$ where $n$ is very large (e.g. 200). Due to the asymetric distribution, $E(Y)$ will be skewed very far towards the extreme upper end of the distribution, dominated by ...
120 views

Variance of sum of $m$ dependent random variables

Let $X_1,X_2,...$ be a sequence of identically distributed and $m$-dependent random variables with $\mathbb{E}[X_i]=0$, $0<Var(X_i)<\infty$ ($m$-dependent means that each $X_i$ is independent of ...
31 views

How to show that there is a number at which the truncated Taylor Series is exact?

So, I'm reading an "easy" proof of the Central Limit Theorem in the book Probability and Mathematical Statistics, by Sahoo, and there is a point where the moment generating function is expanded in a ...