# Tagged Questions

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### Prove Number of Arrivals $N(s)$ up to time $s$ follows $\mathrm{Poisson}(\lambda s)$ Distribution

This comes from my self-study of Durrett's "Essentials of Stochastic Processes" book, page 97. Definition Let $\tau_1,\tau_2,\ldots$ be independent $\mathrm{exponential}(\lambda)$ random variables. ...
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### Rescaling function for probability of $k$ adjacent ones in a binary string

Call $\xi$ a random variable taking values in $\{0, 1\}^{\{0, 1, 2, \ldots, n\}}$, where each character of the string has vaalue $1$ with probability $p$ and $0$ with probability $1-p$ independently. ...
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### Calculation of distribution of a gaussian process

Currently finishing the last year of PhD in statistics, we wonder if you could help us with the following. Let $T = [0,1]$ and $X = \left( X_{t}, t \in T \right)$ be a gaussian process with mean ...
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### Similarity between two probability distribution

I am not sure how to put the question. I am not even sure if this question makes sense at all. I know that the similarity of two discrete (or continuous) distributions can be quantified by ...
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### Probability Density of Convolution of Two Random Processes or Variables

Suppose that we have two stationary random processes $x(t)$ and $y(t)$ with probability density functions $f_{x}(x)$ and $f_{y}(y)$ respectively. Now suppose we form: $z(t) = x(t) \ast y(t)$ What is ...
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### marginal distribution of Ornstein Uhlenbeck process

I am learning the OU process. For now, what I can understand is that the OU process is the strong solution of a SDE $d\sigma²(t)=-\lambda \sigma²(t)dt+dz(\lambda t)$ where z is the compound possion ...
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### Random node observation

The problem is as follows: In a two dimensional plane, nodes are randomly distributed with intensity $\rho$. Each node in the network swings between two states: available, non-avaialable for ...
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### determine type of probability distribution

let us consider following model $$y(t)=A_1 \sin(\omega_1 t+\phi_1) + A_2 \sin(\omega_2 t+\phi_2) + A_3 \sin(\omega_3 t+\phi_3)+ \ldots +A_p \sin(\omega_p t+\phi_p)+z(t)$$ we have three parameter ...
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### Showing the distribution of a poisson process

A large lump of radioactive material has a long half life. Let $D(t)$ be the total number of decays which occur in the radioactive material in the period of $t$ hours starting at noon on a particular ...
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### Probabilistic model of parallel web servers

Note: The following probabilistic model of parallel web servers is abstracted from an engineering project. I am not good at probability theory and I am seeking some evaluations and suggestions. ...