Suppose that $X_1$ ….. $X_n$ are a random sample from a population having an exponential distribution with rate parameter $\lambda$. Use the Central Limit Theorem to show that, for large n, $\sqrt{n}(\lambda\bar{x}-1) \sim Normal(0,1)$ My attempt: honestly I am really not understanding what this question is asking. I can see that for an exponential distribution, […]

It’s always very surprising to learn that some of the entities one has been assiduously studying actually represent negligibly tiny minorities (e.g. continuous functions vis-à-vis all functions)… Now, the Central Limit Theorem, for one, holds only for probability distributions with a finite variance. How common are such distributions in the space of all probability distributions? […]

Note: $\log = \ln$. Suppose $X_1, \dots, X_n \sim \text{Pareto}(\alpha, \beta)$ with $n > \dfrac{2}{\beta}$ are independent. The Pareto$(\alpha, \beta)$ pdf is $$f(x) = \beta\alpha^{\beta}x^{-(\beta +1)}I(x > \alpha)\text{, } \alpha, \beta > 0\text{.}$$ Define $W_n = \dfrac{1}{n}\sum\log(X_i) – \log(X_{(1)})$, with $X_{(1)}$ being the first order statistic. I wish to show $$\sqrt{n}(W_{n}^{-1}-\beta)\overset{d}{\to}\mathcal{N}(0, v^2)$$ as $n \to […]

I saw following exercise from Durrett’s probability theory book and I managed to solve the 1st part, but couldn’t get the 2nd part. Let $X_1, X_2, \dots$ be i.i.d samples with mean $0$, and finite non-zero variance. Denote $S_n = X_1 + X_2+\dots + X_n$. Use central limit theorem and Kolmogorov $0-1$ law to show […]

$X_1, X_2, \dots, X_n$, are $n$ mutually independent r.v.s. $Y_1,\dots,Y_n$ are another set of mutually independent r.v.s. $X_k\mid Y_k=y_k\sim N(y_k,y_k^2)$ and $Y_k\sim\text{uniform}(-k,k)$ for $k=1,\dots,n$. Define $\bar{X}_n=\frac{\sum_{i=1}^nX_i}{n}$ the sample mean of $X_i$’s. I’ve calculated E$[\bar{X}_n]=0$ and Var$(\bar{X}_n)=\frac{n(n+1)(2n+1)}{9n^2}=\frac{(n+1)(2n+1)}{9n}$. How can we prove that $\bar{X}_n$ doesn’t converge to E$[\bar{X}_n]$ in probability? Thanks!

Given a sequence of independent r.v’s $\{X_n\}_{n\geq 1}$ such that $P(X_n=x)=\frac{1}{2}$ if $x=-1$ and/or $x=1$ Let $N\in Po(\lambda)$ be independent of $\{X_n\}_{n\geq 1}$ and we set that $Y=X_1+X_2+\dots+X_N$ One have to show that $\frac{Y}{\sqrt{\lambda}}$ converges in distribution to $N(0,1)$ as $\lambda$ goes to infinity. How can this be shown? I have been thinking of some […]

There are similar posts to this one on stackexchange but none of those seem to actually answer my questions. So consider the CLT in the most common form. Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of i.i.d. random variables with $X_1 \in L^2(P)$ and $\mathbb{E}[X_i]= \mu$ and $\mathbb{V}ar[X_i] = \sigma^2>0$. Denote with $\widehat{X}:= \frac{(X_1+\dots+X_n)}{n}$. Then […]

$X_i(1 \leqslant i \leqslant M)$ is i.i.d distribution of $\mathcal{U}\left({-\pi,\pi}\right)$, $\mathcal{U}$ is uniform distribution. $Y_M = \sum_{i=1}^{M}\cos\left(X_{i}\right)$. When $M$ is large, I want to get a distribution of $Y_M$. Although according to the central limit theory, it turns to be a normal distribution. But based on the simulation result, I think there should be a […]

Ref :Introduction to Mathematical Statistics-Prentice Hall (1994) by Robert V. Hogg, Allen Craig. Now , in the above problem it has been shown that a sequence converges to a random variable X in distribution but the sequence of PMF doesn’t converge to the PMF of X. but we know that “a sequence {Xn} with PDF/PMF […]

I am sorry for the long question! Thanks for taking the time reading the question and for your answers! Context: Let $B_n\sim\text{Binomial(n,p)}$ be the number of successes in $n$ Bernoulli trials of probability $p\in(0,1)$. Let $$\tilde B_n=\frac{B_n-np}{\sqrt{np(1-p)}}$$ be the standardized random variable and let $N\sim\text{N}(0,1)$ have the standardized normal distribution. Let $\epsilon_n(x)$ be the error […]

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