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.
1,589 questions
1 vote
0 answers
49 views
Pass the limit in a sum
I am currently studying the derivation of the distribution of the maximum of the Wiener process ($W$) using the ** Donsker Theorem** (Functional Central Limit Theorem) and the Reflection Principle on ...
- 37
6 votes
1 answer
114 views
Distribution of sum of product [closed]
Let $x_n\overset{p}{\to}c$ and $x_n\overset{d}{\to}N(0,\sigma^2)$ denote convergence in probability to a constant $c$ and convergence in distribution to a random normal variable (with some abuse of ...
- 69
0 votes
1 answer
31 views
MGF of standardized $\bar{X}$ when $X_i$ follows exponential distribution [closed]
When $X_1, X_2, ... ,X_n$ follow an exponential distribution, whose $\theta$ is 2, mgf of $W = \frac{\bar{X}-\mu}{\sigma/\sqrt{n}}$ will be $$\frac{e^{-t\sqrt{n}}}{(1-t/\sqrt{n})^n}$$ I understand the ...
- 39
0 votes
0 answers
19 views
Confidence Interval for Reliability Weighted Samples
I'm trying to do statistical inference on a home poker game. I have calculated the winnings per hour, and I want to create a confidence interval for the variable winnings per hour, in say dollars. The ...
- 2,612
1 vote
0 answers
84 views
A Berry-Esseen-type inequality for uniform distribution
Suppose that $X_1, X_2, \cdots ,X_n$ are i.i.d with $X_i \sim U([-\sqrt3,\sqrt3])$, $\Phi(t)=(2\pi)^{-\frac{1}{2}}\int_{-\infty}^{t}e^{-\frac{x^2}{2}}dx$. Show that there exists $C>0$ such that $$\...
- 11
2 votes
0 answers
126 views
Central limit theorem (Berry-Esseen theorem) for sum of a random number of random variables - from centred to non-centered variables?
The following Berry-Esseen theorem was obtained by Stein's method: Theorem (Chaidee and Keammanee, 2008, Theorem 2.1). Let $X_1, X_2, \dots$ be independent, identically-distributed random variables ...
- 1,108
2 votes
1 answer
125 views
On the convergence of moments in the central limit theorem.
I'm trying to solve a problem from the book Shiryaev A.N. Problems in Probability. Problem 3.4.22. (On the convergence of moments in the central limit theorem.) Let $\xi_1, \xi_2, \ldots$ be any ...
- 81
1 vote
1 answer
57 views
Rearranging expression so that Central Limit Theorem can be applied
I am confused about the following question from a central limit theorem exercise: Does$$\mathbb{P}\left(\frac{1}{\sqrt{n}} \sum_{i=1}^n \left(X_i - \frac{1}{2} \right) > \frac{1}{2} \right) \to \...
