# Tagged Questions

A stochastic process describes evolution of random variables/systems over time and/or space and/or any other index set. It has applications in areas such as econometrics, weather, signal processing, etc. Examples - Gaussian process, Markov Process, etc.

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### proof of Markov chain Monte Carlo

This is the first step of proof of MCMC in my notes I have a question, how come $\pi(x)\pi(x_p\mid x)=\pi(x_p)\pi(x\mid x_p)$? Is it true for any markov chains which are ergodic and aperiodic? The ...
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### Hitting times in two-dimensional case: expectation of Brownian motion at a hitting time

Consider two Brownian motions $$X_{1t}=\mu t+\sigma_1B_{1t}$$ and $$X_{2t}=\mu t+\sigma_2B_{2t}.$$ Here $B_{1t}$ and $B_{2t}$ are uncorrelated. Let $\tau_1$ and $\tau_2$ be the stopping times: \begin{...
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### birth-death process

This is from notes I have 2 questions: why $\lambda h$ or $\mu h$ is the probabiity of one unit change? And let's say the birth rate is 1 per min, and the h is 2 mins, obviously there are two ...
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### Binary Stochastic Programming with Independent or Positively Correlated Co-efficients

A manufacturer can select a maximum of $N$ stores to fulfill orders from a total of $M$ stores who are looking for inventory, $N\le M$. The case when $N\geq M$ is trivially solved when all stores ...
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### Stochastic individual based model

I was going through the article here and can someone please explain what a stochastic individual based model is. Could this be used to model at a population level? Does this model look at each and ...
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### counting process and birth-death process

This is from notes: I have some questions What does it mean by $X(t_2)>X(t_1)$? Let's say there are state i and j and $X(t_2)=i$, $X(t_1)=j$, what does it mean by $i>j$? 2.For the birth-...
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### Brownian motion hitting probability of boundary and going outside

I was solving an exercise which asks the reader to calculate the probability that a Brownian particle $B(t) = (B_1(t),...,B_n(t))$ starting at the origin in $\mathbb{R}^n$ will strike the surface of a ...
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### Statistical comparison between two stochastic algorithms

I am working on a mechanics problem with variability in material properties. For that I need to analyze the efficiency of two methods for same accuracy (confidence level I guess?), the first one being ...
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### What does it mean for a probability density function to have this property?

What does it mean for a probability density function $f(x)$ to have the following property? $$I= 1+\int_{x=0}^{\infty}x^2 \left(\frac{f'(x)^2}{f(x)}-f''(x)\right)dx>0$$ This comes from ...
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### Stochastic vs statistical models

While there are plenty of discussion of the difference between statistics and stochastic, I didn't find a nice explanation for the difference between statistical and stochastic models. Could anyone ...
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### Computing the number of renewals

My question is from the book Introduction to Probability Models, 10th edition, by Sheldon Ross. Page 463, $\S$7.8. There is a paragraph in the book talking about the computation of renewal function: ...
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### What methods can be used for tractable computation of probabilities for evolutionary model of non-independent entities?

I'm trying to extend a simple model which works as follows. We have n 'original' entities which each have a colour. This population evolves by the following events, which occur at exponential rates: ...
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### How do I choose the initial features vectors for a Stochastic Gradient Descent trained SVD++ algorithm?

I'm reading the SVD++ Netflix Recommender Systems paper because I want to be able to properly assess this approach to building a recommender system. How should I choose the initial values of $q_i$ ...
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### Kolmogorov Forward and Backward Equation Intepretation

Let $\lambda_i$ be the sojourn rate of state i, $q_{ij}$ be the transition rate form i to j, and $p_{ij}$ be the transition probability from i to j. The Kolmogorov Forward and backwards equation are ...