# Tagged Questions

1answer
51 views

### Find $a_{n+1}=\frac{a_n^2+1}{2}$ in terms of $n$.

I was trying to prove that for all $n\in \Bbb N$ there are integer numbers $\{a_1,a_2,\ldots,a_n,b_n\}$ s.t. $a_1^2+a_2^2+\dots+a_n^2=b_n^2$. I founded that if $\{a_1,a_2,\ldots,a_n,b_n\}$ have the ...
1answer
79 views

### The normal approximation of Poisson distribution

(I've read the related questions here but found no satisfying answer, as I would prefer a rigorous proof for this because this is a homework problem) Prove: If $X_\alpha$ follows the Poisson ...
2answers
73 views

### Expected values in a sequence

We draw 2 numbers from a normal (gauss) distribution with mean $\mu$ and variance $\sigma$ and we add them to find the first value $a_1$ of a sequence. The second value $a_2$ of this sequence is the ...
1answer
26 views

### Convergence of a series of random elements

Given the normally distribuited random variable $\nu(t)$ with $\mu=0$ and variance $\sigma$, I have to find if the series: $$G(\sigma)=\sum_{k=1}^{\infty}\frac{1}{\exp\left(\nu(k)\right)}$$ where ...
1answer
31 views

### Do greater probabilities approach expected average values with a smaller series?

Let's say you flip a coin 12 times with a goal of getting 6 heads. Then you roll a six sided die 12 times with a goal of getting 2 "ones" faces up. My intuition tells me that rolling the dice has a ...
0answers
99 views

### fractional moments of binomial distribution

I would appreciate your help in learning about the quantity $$\large\sum_{j=0}^J {J \choose j} p^j (1-p)^{J-j} j^\alpha,$$ for any $\alpha > 0$, but in particular for $\alpha\in (0,1)$. Which ...
2answers
58 views

### How to find this limit in statistics?

$$\lim_{n \to \infty} \left( \frac{n}{2}\cdot\left(y - t + n^{-1}\right)\right)$$ where $y$ is a random variable and $t$ is a real number. Its result seems to be infinity but my book said that it ...
1answer
84 views

### Behaviour of Two Coupled Sequences Towards a Stable Distribution

The following question arises from research that I am doing in swarm intelligence. The relationships given come from geometric considerations which, I believe, should not be relevant for this problem. ...
1answer
53 views

### simplify expectation definition Hidden Markov Model

I am reading Rabiner's paper entitled "A tutorial on hidden markov models and selected applications in speech recognition". There is a very simple example where he simplifies the calculation of an ...
1answer
194 views

2answers
421 views

### Infinite sum of random numbers

Let $x_n$, with $n=1,2,\ldots$, be uniformly distributed random variables in $(0,1)$ What is the expected value and probability distribution of the sum $$\sum_{n=1}^\infty x_n^{n^n}$$