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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).

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1answer
18 views

Calculation of Standard Deviation of Sample Mean

This is another statistics problem that I have, which I cannot make sense of: An IQ-test is normal to $ \mu = 100$ and $\sigma = 10$. What is the standard deviation of the sample mean of a sample ...
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1answer
3k views

Proof of the Central Limit Theorem using moment generating functions

Below is a method of proving the Central Limit Theorem using moment generating functions. Let $$X_{1},X_{2},...,X_{n}$$ be a sequence of i.i.d. random variables with expected value and variance $$E(...
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1answer
134 views

Why is the drift parameter of a stochastic process defined in this way?

Within the lecture notes provided for a module on stochastic modeling, the following statement may be found: Let $S_0$ denote the spot price of an asset at time $t=0$ and let $S_\delta^+ = S_0 e^{\...
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0answers
80 views

Where does this equality come from (in this proof of the Central Limit Theorem)?

Can anyone explain to me where the circled inequality comes from in the following proof of the central limit theorem? It says that Lemma 2 is used to obtain this, so I have attached a screenshot of ...
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1answer
95 views

Calculating the probability that one analyst is correct over another

I have a question which goes like this. Two analysts are in dispute about some data they expect to arise in an experiment. In total, they will receive 20 observations. One analyst believes that ...
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0answers
87 views

Equivalent form of Central Limit Theorem

Commonly we use Central Limit Theorem in the following form. Theorem: Let $\{X_i\}_{i=1}^\infty$ be a sequence of i.i.d. random variables such that $E(X_i)=\mu$, $\operatorname{Var}(X_i)=\sigma^2&...
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1answer
488 views

Is the sample mean of a Poisson Distribution Normal? [closed]

I am trying to figure out why we are allowed to use the sample mean in a normal distribution. I tried using moment generating function for $\bar{x}$ to prove that the distribution was normal but ...
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1answer
25 views

Finding the limit of expression used for the central limit theorem

We are some folks working on proving the central limit theorem. The missing piece we lack is finding the limit in the following equation for $n \rightarrow \infty$. Maple is able to find a limit, but ...
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1answer
395 views

Binomial Approximation of Gaussian Distribution

It is said that we may use the binomial coefficients ( a layer from Pascal's triangle) to approximate the 1-D Gaussian kernel with standard deviation $\sigma$, where $\frac{n}{4} = \sigma^2$ and $n$ ...
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1answer
37 views

Find $\lim_{n \rightarrow \infty} [P(X_n>\frac{3}{4}n)+P(X_n>n+2\sqrt{2n})]$

Let $X_n$ be a sequence of independent random variables with $X_n$ having the probability density function of $\chi ^2_{(n)}$(chi-square distribution with $n$ degrees of freedom). Then find $\lim_{n \...
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0answers
202 views

Conditions for Central limit Theorem for compound distribution

Given a compound distribution $$S:=\sum_{k=1}^{N} X_{i}$$ with $N$ is a discrete random variable with values in $\mathbb{N}$ with finite mean and variance. $X_{k}$ are non-negative iid random ...
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2answers
142 views

The Central Limit Theorem and the Scaled Sample Mean

My introductory probability book gives the following two theorems: Theorem 1. For large $n$, the distribution of $\bar{X}_n$ is approximately $N(\mu, \sigma^2/n)$. Theorem 2. The CLT says that ...
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1answer
94 views

Probability of a Sum of Roundoff Errors

Suppose you balance your checkbook by rounding amounts to the nearest dollar. Between 0 and 49 cents, drop the cents; between 50 and 99 cents, drop the cents add a dollar. Find the approximate ...
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1answer
86 views

Deriving Approximate Confidence Interval Using CLT

I'd like to use the central limit theorem that (for large enough n), the random variable $$ Z = \sqrt{n} \left({\bar X \over \mu} -1\right) \sim \mathcal{N}(0,1) $$ and derive an approximate 100(1- $\...
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1answer
79 views

Prove result using Central Limit Theorem

The following is a probability qualifying exam problem that I'm struggling with. Suppose that $X_1,..,X_n$ are i.i.d Rademacher random variables, i.e. $\mathbb{P}(X_i = \pm 1) = \dfrac{1}{2}$....
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2answers
58 views

Central Limit Theorem for a sample

I'm learning basic statistics all by myself so sorry if this is a naive question. Today I was just reading about the Central Limit Theorem. I understand that no matter what distribution a given ...
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0answers
131 views

Chebyshev's Inequality Yields only Conservative Bounds

When talking about the Chebyshev's Inequality $$\Bbb P\left(|\hat{\theta}_n-\theta|\geqslant k\right)\leqslant\dfrac{Var(\hat{\theta}_n)}{k^2},$$ my instructor commented that it gives very ...
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1answer
103 views

Central Limit Theorem Proof using Logarithm Expansion

I am trying to go over a proof of the CLT given by this site. I understood everything up to the point where the expansion for the logarithm was used: $$x=\frac{t^2}{2n}\ +\ \frac{t^3}{3! n^{\frac{3}{...
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1answer
24 views

bayesian hyperpriors central limit theorem analogy

Let's say I have some distribution F, and then put a prior on a parameter of that distribution by using some prior distribution P1. Then let's say I put a hyperprior on that hyperparamter by using a ...
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2answers
315 views

Convergence in Distribution of Sums of Random Variable

Suppose I have $X_1,X_2,...,X_n$ random variables that are independent and identically distributed, from ANY distribution. Suppose that $E(X_i)=\mu$ and $V(X_i)=\sigma^2$. Suppose I define the ...
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1answer
34 views

Limiting Distribution of Unusual Quantities

By central limit theorem, we know that for independent and identically distributed random variables $X_1,X_2,...,X_n$, as $n\rightarrow\infty$, that: $$\frac{\sqrt{n}(\bar X-\mu)}{\sigma}\rightarrow ...
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2answers
306 views

Asymptotic Distribution of Sample Mean

Suppose I have a discrete random variable $X$ which follows a geometric distribution on $x=0,1,2,...$ and I take a random sample from this distribution of size $n$. What is the asymptotic ...
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2answers
23 views

Intercept in linear regression

If i centre the predictor variable by its mean, does this have any effect on the intercept? E.g if I have $$y_i = \alpha + \beta(x_i-\bar{x}) + \epsilon_i$$. See I have centred it above, but does ...
4
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1answer
112 views

The converse on the Central Limit Theorem

The problem is the following: let $X_n$ be i.i.d r.v's and $\{b_n\}$ be a sequence of positive real numbers s.t. $b_n \longrightarrow \infty$ and $$ (I) \qquad \frac{\sum_{k=1}^n X_k }{b_n} \...
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1answer
22 views

Problem when Multiplying Sample Distributions

Question: "An aeroplane is licensed to carry 100 passengers. If the weights of passengers are normally distributed with a mean of 80 kg and a standard deviation of 20 kg.Find the probability ...
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0answers
267 views

Proving Multivariate Central Limit Theorem using Lindeberg Theorem

I'm reading a proof of Multivariate CLT using Lindeberg Theorem. Let $X_n = (X_{ni},... ,X_{nk})$ be independent random vectors all having the same distribution. Suppose that $E[X_{nu}]<\infty$; ...
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0answers
69 views

Convergence of Bernoulli RV to Normal Distribution

Suppose we have $X_1,X_2,...$ Bernoulli RV (not necessarily independent) with $P(X_i=1)=p_i$. I am trying to understand what conditions are sufficient in order for a CLT to hold. Surely if the ...
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0answers
50 views

Central Limit Theorem of an expectancy

Let $S_n(\psi)$ be a uniformly bounded statistic that depends on a real-valued random variable $\psi$, whose the distribution $F_{\psi}$ is known. I established using the Central Limit Theorem that ...
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1answer
69 views

Convergence in distribution, CLT (solution check)

$(X_n), (Y_n)$ independent sequences of iid random variables such that $EX_1=EY_1=0$, $\operatorname{Var}(X_1)=\operatorname{Var}(X_2)=\sigma^2$. $S_n=\sum_{k=1}^nX_k$, $T_n=\sum_{k=1}^nY_k$. 1) ...
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0answers
49 views

Determine limit distribution

Let $X_1,\dots,X_n$ be an iid sample from a uniform distribution $U[-\theta,\theta]$ for some parameter $\theta>0$. Define $X_{(n)}=\max\{X_i: 1\leq i\leq n\}$ and to estimate $\theta$ consider the ...
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1answer
146 views

Applying central limit theorem to show convergence in distribution

Let $X_1, X_2, \cdots$ be i.i.d random variables with mean $0$ and finite variable $\sigma^2$. Use the central limit theorem and Slutsky's Theorem to show that $\frac{\sum_{m=1}^{n} X_m}{\sqrt{\sum_{m=...
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0answers
134 views

Trouble understanding the Lindeberg-Feller Central Limit Theorem

I am having trouble understanding the notation in the following Lindeberg-Feller CLT. The notation isn't defined anywhere else in the text so I would like to clarify a few things. Firstly, what is $...
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1answer
162 views

Central Limit Theorem and Sum of Random Variables

Let each $X_i$ be i.i.d and $Y_i=\ln(X_i)$ is exponentially distributed with mean $\alpha^{-1}$ and variance $\alpha^{-2}$. Now let $$S_n=\left[ \prod_{i=1}^{n}X_i \right ]^{n^{-1}}.$$ Use the ...
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1answer
31 views

Two random variables generated with common random varibales

I have two random variables (RVs) (in general I have more $U_1,\cdots,U_N$) which are defined as $$U_1=\sum_{i=1,i\neq 1}^{N}x_i$$ $$U_2=\sum_{i=1,i\neq 2}^{N}x_i$$ where $x_i$s are i.i.d. RVs. Since $...
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0answers
61 views

Convergence of ratio of population mean and sample of sample means to a fixed constant

Let $A = \lbrace a_1, ..., a_n \rbrace$ be a set of numbers in $\Bbb R$. Let $\omega = \lbrace x_i \rbrace_{i=1}^k$ be a random sample of $k$ distinct elements in $A$. Let $\bar{\omega} = \frac{\sum_{...
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1answer
104 views

Help verifying the Lindeberg condition

Suppose $X_1, X_2,...$ is a sequence of independent, identically distributed random variables with $\mathbb{E}[X_n] = 0$ and $\mathbb{E}[X_n^2] = 1$. Find sequences of constants $a_n$ and $b_n$ ...
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1answer
123 views

Central limit theorem with different variance

Let $\{ X_n, n\geq 1 \}$ be a sequence of independent random variables with $P(X_n=n^{\lambda})=P(X_n=-n^{\lambda})=\frac{1}{2}$ where $0<\lambda<\infty$ Show that the central limit theorem is ...
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3answers
198 views

Lindeberg condition fails, but a CLT still applies

I'm having difficulty with this old qualifying exam problem. Suppose we have a sequence of independent R.V's $\{X_n\}_{n\in\mathbb{N}}$ satisfying, $$ \mathbb{P}(X_n = \pm n^2) = \frac{1}{12n^2}, \;\...
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2answers
403 views

Chebyshev's inequality and CLT to approximate 1.000.000 coin tosses probability

Let S be the number of heads in 1,000,000 tosses of a fair coin. Use Chebyshev’s inequality, and the CLT to estimate the probability that S lies between 499,500 and 500,500. Use the same two methods ...
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1answer
208 views

Probability to have exactly 55 heads on 100 coin flips and CLT

What is the probability to have exactly 55 heads out of 100 coin flips ? The exact answer is $100\choose{55}$$\frac{1}{2^{100}} \approx 0.0484$ We can see this game as a repetition of Bernoulli($\...
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0answers
35 views

Example of easy calculations with the central limit theorem in higher dimensions

Let $X_1,X_2,\dots \,$ i.i.d. random variables with values in $\mathbb R^2$ and let $S_n:=\sum_{i=1}^n X_i, n\in\mathbb N. $ $U_A $ denotes the uniform distribution on A. Calculate the weak limes of $\...
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1answer
159 views

Example of central limit theorem fail due to dependence (for tuition)

I want to get better understanding why the central limit theorem can fail. I have seen examples that the theorem fails when the variables are not identically distributed, and when the variance is not ...
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2answers
707 views

Central limit theorem - Coin toss

We toss $n=200$ Euro coins on the table. I want to calculate, using the central limit theorem, the probability that at least $110$ coins have tutned on the same side. $$$$ Do we have to consider ...
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0answers
106 views

On the rate of convergence of the central limit theorem

As I have seen, there are many questions on this site concerning the rate of convergence of the central limit theorem: for example this, this, this, and this. In particular, the third question of this ...
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1answer
60 views

CLT applicability [closed]

Let $\xi_1, \xi_2...$ be i.i.d. random variables. $\mathbb{E}\xi_i = 0, \mathbb{E}\xi^2_i = 1 \ \forall i $. Let $\lambda_1, \lambda_2, ...$ be a sequence of real number such that $$ \frac{\max_{k \...
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1answer
97 views

Another application of the Central Limit Theorem

Let $Y_1,Y_2,\dots$ i.i.d random variables with values in $\mathbb R^2$. Let $Z_n= \sum_{i=1}^n Y_i, n\in\mathbb N$. I want to calculate the weak limit $\frac {Z_n}{\sqrt n}$ if $Y_1\sim U\{(-1,0),(1,...
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1answer
318 views

Calculation problem with Central limit theorem

Let $X_1,X_2,\dots\,$ i.i.d random variables with mean zero and variance $1$. Let $S_n=\sum_{i=1}^n X_i\,,n\in \mathbb N.$ Compute the weak limes $\lim_{n\to\infty} \frac1n \sum_{i=1}^n \frac{S_i}{\...
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1answer
440 views

Functional Central Limit Theorem

My question is related to the definitions of Functional Central Limit Theorem mentioned on Wikipedia Version 1: If $X_1,X_2,X_3,\dots,X_n$ are iid random variables with mean 0 and variance 1, then $...
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1answer
47 views

Heads/Tails 900 times, find k such that $P(k \leq T \leq 480) = 0.1$ with T the number of Tails

Heads/Tails 900 times, find k such that $P(k \leq T \leq 480) = 0.1$ with T the number of Tails. I will use the CLT because I know that $$P(z_{1} \leq \frac{S_{N} - \mu N}{\sqrt{N} \sigma} \leq z_{2})$...
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1answer
85 views

How To find EXi Central Limit

The problem: ...