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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.

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General Poisson process

Define a generalised Poisson process as an arrival process that begins at time 0 and that satisfies: The independence property: the number of arrivals during two non-overlapping intervals are independent. Small interval…
user40333
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Are the two definitions of branching proceses equivalent?

From Wikipedia The most common formulation of a branching process is that of the Galton–Watson process. Let $Z_n$ denote the state in period $n$ (often interpreted as the size of generation $n$), and let $X_{n,i}$ be a random variable denoting the…
Tim
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Discrete random variable with probability generating function problem. Help!

Suppose $X$ is a discrete random variable with probability generating function: $G_X(\theta)$ = $2(3-\theta)^{-1}$ 1) If $Y$ = $X^{2}$ write down $P(Y=k)$ for $0\leq k \leq 10$, and find $E(Y)$ Firstly i know i have to get $G_X'(\theta)$ but i don't…
water723
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Simple Random Walk, Optimal Stopping Strategy

Consider a simple random walk on the four vertex graph (Square shape with A,B,C,D being vertex) Assume that the payoff function is: $$f(A) =2,\\ f(B) = 4,\\ f(C) = 5,\\ f(D) = 3.$$ Assume that there is no cost associate with moving, but there is a…
James
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How can I calculate the probability that in 10 days died at least 160 cells, with lambda 15 per day

Suppose that cell death occurs according to a Poisson process with rate lambda = 15 per day. Calculate the probability that after 10 days have died least 160 cells. I am very confused, I dont know to solve this. I know…
Salvattore
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Conditional expectation equality

Does this statement hold and how to prove it correctly? $$ \mathbb{E}(\mathbb{E}(X\mid \mathbb{F})^2) = \mathbb{E}(X\mathbb{E}(X\mid \mathbb{F})) $$ Any help? Thanks. $\mathbb{F}$ is a sigma algebra. $X$ may or may not be $\mathbb{F}$-measurable.
luka5z
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Gaussian process property

Show that for a Gaussian process $z$ with zero mean we have $=e^{/2}$. If we denote by $f_G(z)$ the Gaussian process…
Jesús Ros
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How to generate a series of points in a convex set?

For an arbitrary convex set, may be a polyhedron or an ellipsoid how can I generate N uniformly distributed points inside?
user111333
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A question about predictable stochastic process.

Let $X_t$ be predictable with respect to filtration $(\mathscr{F}_t:t\in[0,T] )$. If I observe the process over an interval $[0,s],0
Andrey Carl
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Distribution F in a renewal process

Referring to the Ross textbook stochastic processes, define the inter arrival times $x_i$ follow a distribution $F$. What is F representing here the PDF or CDF? What would $\overline{F}$ represent?
knk
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Variance and covariance function of stochastic process

Let $X_t = Z_t + \theta Z_{t-1}$ where $ \left\{ Z_t \right\} \approx WN(0, \sigma^2)$. Find variance $ VarX_t$ and covariance function. Of course we have $EX_t = 0$. Then $VarX_t = EX_t^2 = E(Z_t^2 + 2 \theta Z_t Z_{t-1} + \theta^2Z_{t-1}^2) =…
Thomas
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Stochastic Process vs a sequence of r.v.

A stochastic process $\{X(t) : t \in T \}$ is a collection of random variables. That is for each $t \in T$, $X(t)$ is a random variable. [Sheldon Ross] I am trying to understand the "collection" part with a sequence of random variables $\{X_n…
Lemon
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jump variation of square integrable martingale.

I'm studying Stochastic calculus and applications, Cohen. He claims that As $M \in \mathcal{H}^{2,d}$, we know $\lim_k \sum_{j=1}^k \mathbb{E}[(\Delta M_{T_n})^2] < \infty$. Note that $T_n$ is all jump times of $M$, which is either totally…
Han
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Modeling repeated coin tosses

I wanted to confirm if the theoretical form outlined below is inline with expectation for a model. Say we have a coin that is tossed with probability $p$. We are interested in arriving at the pattern T,T,H after continuous tosses, and would like to…
Will_E
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Upscale Probability Mass Function of a Discrete Random Variable

I'm currently struggling with a problem that sounded rather easy to me at first sight, yet I was not able to obtain a working solution: I have a random variable that describes how often an event occurs in a given time range $t$. The probability mass…
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