Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [stochastic-processes]

For questions about stochastic processes, for example random walks and Brownian motion.

A stochastic process is a collection of random variables representing the evolution of a system of random variables over time. A typical example is a .

A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

16128 questions
0
votes
1 answer

Double checking solution (Poisson process)

I am just double checking to see if my solutions are correct for these two questions. Assume that for both customers arrive at rate lambda. I am unsure about the second one so confirmation would be great. Thanks! 1) Amy is the first customer to…
icobes
  • 1,109
0
votes
0 answers

Generalized stochastic equation

We study the diffusion of a message (say, like a tweet) on a social network. To this end, we use the following simplified model. Let $X_n$ be the number of individuals that have received the message at time $n$, then assume that the $k^{th}$…
New comer
  • 11
0
votes
1 answer

Stochastic Process Simulation

I am trying to understand some basic stochastic process simulation for population growth. Let us say, a population grows exponentially at a constant rate $R$, given by the equation, $$ P(t+1) = R P(t) $$ In the book "A Biologist's Guide to…
user2167741
  • 463
0
votes
1 answer

Understanding Stochastic process transition probabilties

I'm trying to understand the statement: The probability in going form state $i$ to state $j$ in $n$ steps is given by $$\left(P^n\right)_{i,j}$$ this is followed by an example for $k=2$ steps: $$P_{i,1}P_{1,j} + P_{i,2}P_{2,j} + ... +…
atomsmasher
  • 324
0
votes
0 answers

Branching Process - Extinction Probability

I'm studying the following example but unable to understand a few things: Assume all particles act independently of each other, and the probability that a particle produces $k$ offsprings is the same for all particles. Denote this probability with…
tkj80
  • 347
0
votes
1 answer

Inter-arrival Time of Poisson Process including $t > 0$

Given a Poisson Stochastic Process with parameter $\lambda$, determine the the distribution of the inter-arrival time given that it contains $\tau > 0$ (it is not exponential). My approach I want to calculate $$\Pr [X(t) = 1 | X(\tau) = 0],…
bolzano
  • 1,032
  • 6
  • 17
0
votes
1 answer

Help with summation stochastic

I don't understand how to get from the first line to the last. What happens to the summation? Please help! An important piece of information given in the question is that the transition probability from state 0 to state 1 is equal to 1. N.B. o(h) is…
icobes
  • 1,109
0
votes
1 answer

Pure Birth Process Probabilities

I have no clue how to proceed with this question. Please help me in deriving the differential equations! Thanks very much... I really appreciate it :)
icobes
  • 1,109
0
votes
2 answers

Pure Birth Process Question

I would appreciate any possible help for this question because I have no clue what to do! Thanks so much! Consider a population made of a fixed number (N) of people. At time t=0 there is only one infected individual and N-1 susceptible people in the…
icobes
  • 1,109
0
votes
0 answers

Conditional expectation with respect to Sub-σ-algebra

Given a random variable $X$ on the space $\Omega$ endowed with sigma algebra $\mathcal{A}$. Let $\mathcal{F} \subset \mathcal{A}$ be a sub-sigma algebra. How to prove that for the expectation value we have $$E[E[X∣\mathcal{F}]]=E[X]$$?
user267839
  • 7,293
0
votes
2 answers

Having trouble with Exponential distribution

I'm working on the following sample problem and there's no solution for it. Let X be the amount of time a student needed to finish a midterm. Assume that X follows an exponential distribution with mean 30. Compute V(X) Compute P(X>=45) What is the…
echo
  • 453
0
votes
1 answer

Waiting system, with Poisson and Exponential distributions, and maximum waiting time

Consider a single server queuing system, where customers arrive according to a Poisson proccess with rate $\lambda$, service times are exponential with rate $\alpha$. When an individual comes into the system, there is a person on the waiting line…
JPC
  • 1
0
votes
1 answer

Expected time involving a Poisson process

A chicken wishes to cross a single lane of traffic on a long-straight road. The arrival process of cars is a Poisson process of rate $\lambda$. She needs a car-free interval of length $c$ in order to safely cross the road. How long does it take her…
Btzzzz
  • 1,113
0
votes
1 answer

Can we always find out the equilibrium distribution of a Markov chain

I am confused about whether the equilibrium distribution can always be calculated of a Markov Chain, if it is not, then in what situation can the equilibrium distribution be found out? Also, for example, if an equilibrium distribution is $\pi =…
Kyle
  • 87
0
votes
1 answer

Expectation value of branching process

Let $\mu$ and $\sigma^2$ be the mean and variance of the number of offspring of one individual. $\mu=\Sigma_{j=0}^{\infty}jP_j$ We want $E[X_n]$ The note is as follow: Note that $X_n = \Sigma_{i=1}^{n-1}Z_i$,where $Z_i$ is the number of offspring…
Kyle
  • 87
1 2 3
…
46 47