# Questions tagged [probability-limit-theorems]

For question about limit theorems of probability theory, like the law of large numbers, central limit theorem or the law of iterated logarithm.

1,126 questions
Filter by
Sorted by
Tagged with
49 views

### 1) Is there other way to provide a proof WITHOUT using the theorem about order of series? 2) Changing the hypothesis how can I solve this?

QUESTION: Let $X_1, X_2, \cdots$ be independents random variables such that $P(X_n=-n^{\theta})=P(X_n=n^{\theta})=1/2$. If $\theta > -1/2$ prove that the Lyapunov condition works and this sequence ...
45 views

36 views

### Calculating probability limit

I tried taking conditional probability on $\epsilon,$ to change the question in a form where we are taking plim of $\mu^2$ plus some noise. However, I'm having difficulties showing the noise part ...
33 views

### Probability convergence / Convergencia [closed]

I have to prove the almost sure convergence. I really do not know how to start this, i have been struggling with this all the day and i am not going anywhere. The $X_1, X_2,...$ are random variables ...
14 views

### A non- trivial algorithm to estimate the log-exp expected value?

While working on a project, I am facing the an expression in the form $$\frac1N\log\mathbb{E}_{\textbf{X}}e^{\frac12\textbf{X}^TA\textbf{X}}$$ for $\textbf{X}$ uniformly distributed over $\{\pm1\}^N$. ...
33 views

59 views

37 views

### Calculating Upper Bounds for Probability when Extrapolating

This question is from the MIT 6.042J 2005 Final exam. The problem asks us to find an upper bound on the probability that a program runtime will be $\geq60$ seconds given that the expected runtime is ...
150 views

### Aproximation of the Normal Distribution by the Normal Density Function

In Feller's introduction to probability the next lemma is stated: "As $x\rightarrow \infty$ $\tag1 \frac{1-R(x)}{x^{-1}n(x)} \rightarrow 1$ Where $R(x)$ is the normal distribution and $n(x)$ is ...
25 views

### How to determine the expected distribution of the population?

I have a sample with a size n = 30. I know the mean and standard deviation of the sample. I also know the distribution of the sample which is normal. With this data, is it possible to determine the ...
70 views

### A probability question, need help.

Suppose we have $N$ people, $N\in\mathbb{Z}_+$. These people arrive at a place at a random time $t_1, t_2, \dots, t_N$, where $t_k\in(0,1)$, following certain distribution (it is unknown in my case, ...
66 views

113 views

21 views

60 views

### Bounded random variables $X,Y$ satisfying $\mathbb{E}(X^mY^n) = \mathbb{E}(X^m)\mathbb{E}(Y^n)$ for every $m, n\in\mathbb{N}$ are independent

Suppose $X, Y$ are bounded random variables and we have that for every $m, n$ positive integers, $\mathbb{E}[X^mY^n] = \mathbb{E}[X^m]\mathbb{E}[Y^n]$. Then show that $X, Y$ are independent. I have ...
117 views

### Slowpoke and Doubles probability — does it converge as the numbers of laps increases?

I posted this on puzzling stack exchange three months ago and it was immediately closed as off-topic, for being a "fairly straightforward probability calculation". I have the solution to the ...
50 views

### Almost sure convergence in parameters preserves convergence in distribution

Let $X_n$, $n\in\mathbb{N}$ denote a sequence of real-valued random variables that converges in distribution to the standard normal distribution. In addition, each $X_n(c)$ is a function of a real-...
35 views

35 views

### Is there any method to solve this probability and find an closed form expression?

Suppose I have Shannon's rate equation $$R = \sum_{i=1}^{N}\log\left(1+\frac{P\gamma_i}{\sigma^2}\right).$$ Here $N$ is the number of OFDMA subcarriers, and $\gamma_i$ are channel coefficients that ...