1
vote
0answers
33 views

Law and Brownian Bridge

Let $Z_{t}= W_{t}-tW_{1}$ and $Y_{1}=\sup_{0\leq t\leq 1}Z_{t}$, $(W_t, t \geq 0)$ standard Brownian motion Find the law of $Y_{1}$ I know that $\textbf{P}(\sup_{0\leq t\leq 1}W_{t}\geq x , ...
1
vote
1answer
55 views

Exponential of Squared Brownian Motion

Long time lurker, first time posting! Have a problem, that looks familiar but I can't put my finger on it. Need to calculate $\mathbb{E} [\exp(aW_T^2)|F_t]$ where $W_t$ is an $F_t$ adapted standard ...
3
votes
1answer
56 views

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. ...
0
votes
0answers
12 views

Error of a Serial Processs

Give random variable X and two processes A, B . Assume that $ Y_{1}, Y_{2}$ are estimated versions of X by using processes A, B respectively, with probability: $P\left \{ \left | X-Y_{1} \right ...
2
votes
1answer
30 views

derivation law from the call option formula

i am reading a article about the option pricing. and i got stuck with some typical statement. $C(K)=\int (x-K)^+\mu(dx)$ is given. here, $\mu$ is implied law of asset price and C(K) is the price ...
0
votes
1answer
35 views

Measurability and knowledge

there seems to be a subtle relationship between knowledge and measurability. If I have a stochastic process $(X_n)_n$, then for example a stopping time ( other examples would be martingales, ...
0
votes
1answer
36 views

How to prove that convergence in MGF implies Convergence in Distribution?

I know that if the moment generating function of two distribution converges to the same function then the two distribution converges in CDF. But how can we prove this thing explicitly ?
22
votes
1answer
393 views

Zombie outbreak on a $k$-regular graph

Suppose we have a zombie outbreak on a connected $k$-regular graph of order $n$. There are $n_0$ initially infected zombie nodes, and each turn, each zombie infects its neighbors with probability ...
0
votes
1answer
15 views

Is squared Brownian Motion a gaussian process?

I am working at the following SP, given by $(X_t)_{t\geq0} = \alpha W_t^2+\beta t$ where $W_t$ is Brownian motion and $\alpha,\beta$ real. I managed to calculate mean and covariance function and now I ...
1
vote
0answers
32 views

Convergence of probability density functions

Assume that a sequence of random variables, $(X_t)_{t\geq 0}$, converges in distribution to a random variable $X_0$, as $t\to 0$. Also assume that $X_t$ and $X_0$ have $C^{\infty}$-probability density ...
0
votes
0answers
19 views

Transition matrix in left-right hidden semi-Markov model

I'm developing a hidden semi-Markov model left-right . In a left-right model a sequence of $M$ states starts in state $1$ and ends in state $M$, with no repetition of states. Since the model is ...
0
votes
1answer
28 views

number of ones with neighbours in a random binary string

Consider a sequence of i.i.d. random variables $(\xi_i)_{1 \leq i \leq L}$ such that $\xi_1 \in \{0,1\}$ and $P(\xi=1)=p$. Introduce the function $N : \{0,1\}^{L} \rightarrow \mathbb{N}$ which counts ...
3
votes
1answer
30 views

What is the probability that $k$ events have occurred at time $t$, i.e., $\Pr[N(t)=k]$?

Assume that the starting time $T_0=0$. There are $n$ events that occur sequentially at time $T_1$, $T_2$, …, $T_n$, ($T_k\geqslant T_{k-1}$). Suppose the time intervals $\Delta{T_k}\,(\Delta{T_k} = ...
1
vote
1answer
34 views

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. ...
2
votes
1answer
48 views

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 ...
0
votes
3answers
100 views

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 ...
1
vote
1answer
38 views

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 ...
0
votes
0answers
32 views

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 ...
0
votes
1answer
18 views

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 ...
3
votes
0answers
41 views

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 ...
1
vote
1answer
25 views

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 ...
0
votes
0answers
39 views

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. ...
1
vote
0answers
23 views

Waiting time probability question

I want to solve the following problem: A dentist works 4 hours a day. Patients arrive on the average of 1 per 20 minutes and one patient spends on average 15 minutes with the dentist. Both time ...
0
votes
0answers
22 views

Distribution and Laplace transform

I'm having trouble understanding this problem from Resnick's Adventures in Stochastic Processes: The problem says: Suppose $F$ is a distribution of a positive random variable and $p_k \geq 0, ...
1
vote
1answer
34 views

relation between multivariate probability generating function and univariate ones

Suppose I have two independent integer random variables $X_1$, $X_2$ (with constraint that $X_1+X_2\le N,0\le X_1\le N,0\le X_2\le N$), with probability generating functions $g_1(z)$, $g_2(z)$. Now I ...
0
votes
0answers
15 views

Is reflected levy process a feller process?

In some literature , there is a concept similar to reflected Brownian process. Assume that $L_{t}$ is a levy process (may be we can assume it's not a Poisson process) then reflected Levy process ...
0
votes
1answer
36 views

multivariate probability generating function

Suppose I have three random variables $X_1$, $X_2$ and $X_3$, with probability generating functions $g_1(z)$, $g_2(z)$ and $g_3(z)$. Now I have a joint-distribution $P(X_1-X_2,X_1-X_3)$, whose ...
1
vote
1answer
98 views

Probability of Renewal Processes

Suppose that there are two brands of replacement components, Brand X and Brand Y, and that for political reasons a company buys a replacements of both types. When a Brand X component fails it is ...
2
votes
2answers
155 views

Traversing an array and counting the number of distanct number from the given elements in an array.

You are given an array $A[0 \ldots n-1]$ of $n$ numbers. Let $d$ be the number of \emph{distinct} numbers that occur in this array. For each $i$ with $0 \leq i \leq n-1$, let $N_i$ be the number of ...
2
votes
0answers
46 views

A Lemma in the book “ Mathematical Method for financial markets” (Chapter 5, Section 5.7)

In page 307, Section 5.7, Chapter 5 of the book "mathematical methods for financial markets" by Jeanblanc, Yor and Chesney, Lemma 5.7.1 is given as follows: Lemma 5.7.1.1 Let $W$ be a Brownian ...
2
votes
2answers
133 views

Professor has 4 umbrellas, Markov chain and Probability

OK this problem is making me tear my hair out. I need someone to walk me through this in baby-steps method like 1 + 1 = 2. I am trying to figure out what I don't understand. I know this is going to be ...
0
votes
0answers
28 views

Learning resources for Probability Distributions/Models

I've a good background in basic probability. I need to learn and get a good grip on the probability distributions and stochastic processes, counting processes, and other related topics. I am already ...
1
vote
2answers
78 views

How to prove two stochastic processes have the same distribution

Let $C([0,\infty), R)$ be the canonical space of continuous functions. Assume $(\Omega, \mathcal{F}, \{\mathcal{F}_{t}\}_{t\geq 0})$ is a measurable space with a filtration. Let $P, Q$ be two ...
5
votes
1answer
186 views

Asymptotics of sum of binomial distributions

Definition 1: For any random variable $X$, we define $\mathrm{Bin}(p,X)$ as a variable with binomial distribution having parameters $p$ and $X$. Definition 2: For all $i \in \mathbb{N}$, define ...
1
vote
0answers
23 views

The sign of pure jump Levy process

Suppose $(\Omega, \mathcal{F}, P)$ is a probability space. Assume $(X_{t}, P)$ is a Levy process with generating triplet $( 0, 0, \nu)$ with $X_{0}=0$. This means there is no continuous part in ...
-1
votes
2answers
62 views

What is the probability that a student knows the answer given that he has answered it correctly? [closed]

A large class in stochastic processes at a school is taking a multiple choice test. For one particular question with m proposed multiple choice answers, the fraction of students who know the answer is ...
0
votes
0answers
61 views

Probability distribution after n-steps with different initiation state in Markov chain

The transition matrix at n-th time step for a discrete time Markov chain with $ S = \{1, 2, 3, 4\} $is given as below: $$ P(n) = \pmatrix{0 & 0.6 & 0.4 & 0 \\ 0.8 & 0 & 0 & ...
1
vote
1answer
58 views

Expectation and covariance of a gamma distribution.

Assume that the conditional distribution of $U$, given $L$ is uniform over the interval $[0,L]$ and $L$ itself has the gamma-distribution with the density described below. \begin{equation} ...
1
vote
2answers
175 views

Prove that integral is a Gaussian random variable, compute its mean and variance

I have to prove that $X_t=\int_0^t W_s ds$ is a Gaussian random variable. I need also to compute it's mean and variance. My attempt: Let $W_t$ be a simple adapted process ...
2
votes
1answer
54 views

Deriving statistical distributions from games

The normal distribution can be derived from basic principles and calculus The Normal Distribution: A derivation from basic principles. Are there other distributions that can be derived like this from ...
0
votes
0answers
15 views

Particles arriving, emitting fields, and leaving

I have a field emitted from a particle at some position $c$, with an intensity distribution given by $I(x,y,z) = t*((\Delta x)^3 + (\Delta y)^3 + (\Delta z)^3)^{\frac{1}{2}}$, where $t$ is the time ...
2
votes
0answers
45 views

Change probability distribution such that output symbols change minimally

I'm not a mathematician, so my explanation will be slow and plodding. Briefly, I want to create a series of discrete probability distrubution functions, that will generate output symbols in the ...
0
votes
1answer
32 views

Distribution function for a fair cube

I have a fair cube with n-sides (1,...,n). I have a random variable R which is the maximum number of the cube that will appear after dicing the cube k-times. What is the distribution function of R? ...
1
vote
1answer
46 views

How to show that two random proceses have the same family of finite dimensional distributions?

I got two random processes: $$y_t=e_t-\frac{1}{3}e_{t-1},\ e_t\sim\mathcal{N}(0,9)\ \text{i.i.d.}$$ $$y_t=e_t-3e_{t-1},\ e_t\sim\mathcal{N}(0,1)\ \text{i.i.d.}$$ I want to show that both have the ...
2
votes
0answers
149 views

Comparing the stopping times of two stochastic processes

Let $f_1$, $f_0$, $g_1$, $g_0$ be $4$ distinct density functions on the real numbers $\mathbb{R}$ with the corresponding distribution functions $F_1$, $F_0$, $G_1$, and $G_0$, respectively. The ...
1
vote
1answer
75 views

Question about Dirichlet process

Let $\varpi$ be a Dirichlet process on $[0,1]$ with concentration parameter $\varepsilon$ and base measure $\alpha$, where $\alpha$ is a Beta distribution with parameters $\alpha_0$ and $\alpha_1$. ...
1
vote
0answers
34 views

Next event prediction

I've got a list of a number of events and the time they've happened. All the time stamps are in half hours: ...
0
votes
3answers
52 views

Get one of the two random variables's distribution function from limitation [duplicate]

This is a very fundamental problem. In the Stochastic Processes textbook, it says that: The Continuity Theorem of Probability allows us to conclude that $$F_X(x)=\lim_{y \to \infty}F_{XY}(x,y)$$ ...
1
vote
0answers
188 views

Stationary Increments of a Poisson process

Let $\{N(t),t\geq0\}$ be a Poisson process, i.e. for $t\geq0$ and $n\geq0$, $P(N(t)=n) = \dfrac{e^{-\lambda t}(\lambda t)^n}{n!}$, with $\lambda>0$ a constant. Prove that $$P(N(t+s)-N(t) = n) = ...
0
votes
0answers
41 views

Probability distribution of distances between randomly elected numbers?

Assume that a list is generated as follows. Each natural number is included in the list with a given probability p. Let G be the gap between two numbers (gap 1 is zero to number 1 gap two is distance ...