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).

820 questions
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Normally distributed experiment: argue that this is (is not) true

I have defined a set of graphs, where each graph of the set fulfill the following conditions has $n$ nodes degree is fixed to $d$ The edges are distributed in a way such that there are no bridges (...
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Probability space of random variable according to central limit theorem

According to Central Limit Theorem: $\Omega$ = {$\omega_1$, $\omega_2$,..., $\omega_n$}, $Event$, $p$, $X$ : $\Omega$ $\rightarrow$ ${\rm I\!R}$ $n$ elements $e_1$,...,$e_n$ are drawn ...
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Central limit theorem/ poisson distribution

Let$\ X_1,X_2,...,X_n$ be independent Poisson random variables with parameter$\ λ=1$, use the Central Limit Theorem to prove: $\ \lim_{n→∞} \frac{1}{e^n} \sum_{k=0}^n \frac{n^k}{k!} =\frac{1}{2}$ My ...
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Approximation error from using the Central Limit Theorem

A bank accepts rolls of pennies and gives 50 cents credit to a customer without counting the contents. Assume that a roll contains 49 pennies 30 percent of the time, 50 pennies 60 percent of the ...
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Any central limit theorem-like theorem stating distribution of something will be convergent to uniform distribution?

The central Limit Theorem states that the distribution of the sum of independent random variables (sampled from any population) converges to a normal distribution. Are there any similar theorems for ...
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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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Central Limit Theorem and Law of Large Numbers for stable linear stochastic systems

I am new to Markov chain in continuous state space. It would be great if anyone can help me with this. Consider a linear stochastic system: $x_{t+1}=Ax_t +w_t$, where $w_t$ i.i.d. Gaussian with zero ...
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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: ...
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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 ...
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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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Central limit theorem example

I am not able to make progress with the following exercise. Assignment: The advertising board is lit by one 100W bulb. Using the central limit theorem determine the minimum number of bulbs needed ...
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Explain why a gamma random variable with parameters $(t, \lambda)$ has an approximately normal distribution when $t$ is large.

Explain why a gamma random variable with parameters $(t, \lambda)$ has an approximately normal distribution when $t$ is large. What I have come up with so far is: Let $X=$ the sum of all $X_i$, ...
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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 ...
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When to use the continuity correction for normal approximations of binomial probabilities.

so I'm confused as to when you actually use continuity correction. If a problem deals with a binomial distribution and we are asked to find probabilities using normal approximation (provided np>5 and ...
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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 ...
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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 ...
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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 ...
The question is: $g:R\rightarrow R$ has at least three bounded continuous derivatives and let $X_i$ be $iid$ and in $L^2$. Prove that: i) \$\sqrt{n}[g(\overline{X_n}) - g(\mu)]\xrightarrow{w} N(0,g^{...