1
vote
1answer
14 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
1answer
24 views

Computing joint probability [on hold]

Let $X,Y\sim \text{Exp}(1)$ (exponential random variables with parameter $1$). Then prove that $$Pr(X> z_1, \frac{Y}{X} > z_2) = \dfrac{e^{-z_1 (1+z_2)}}{1+z_2}, \forall z_1,z_2>0$$
0
votes
0answers
24 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. ...
0
votes
0answers
14 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
19 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, ...
2
votes
1answer
31 views

Convergence of Quantiles moments.

QUESTION: Let $F$ be an absolutely continuous distribution function whith density f, and $F_{n}$ be its nth empirical distribution. Suppose that $t\in (0,1)$ is constant. Is true the convergence ...
1
vote
1answer
29 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
9 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
30 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
65 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
145 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
31 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
73 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
23 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
49 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 ...
4
votes
0answers
131 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
18 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
51 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
34 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
36 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
111 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
42 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
13 views

Values of Cells in a 2D Grid - Evaluating Distribution

A function iteratively executes on a 2d grid. Each cell in the grid has a (decimal) value associated with it. A function input parameter impacts the distribution of values in the grid. For example, ...
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
24 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
29 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
42 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
147 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
69 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
33 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
120 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
37 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 ...
1
vote
1answer
83 views

Autocovariance function of a Poisson process transformation

here is the problem formulation: Let $\{N_t,t \ge 0\}$ follow a Poisson process with rate parameter $\lambda$ and let $A$ be a random variable with zero mean and unit variance, $A$ is independent of ...
1
vote
0answers
11 views

Distribution properties of the arithmetic mean of a subordinated Gaussian Process

Let $(X_i)_{i\in\mathbb{N}}$ be a stationary standard gaussian sequence and $G$ be a function such that $\operatorname{E}\left[G(X_1)\right]=0$ and $\operatorname{Var}(G(X_1))<\infty$. Further let ...
0
votes
1answer
40 views

Where is the error? Expectation, independent random variables

Let $X,Z$ be two correlated variables and $Y,Z\sim N(0,1)$ where Y is independent of $X,Z$. Consider the expectation: $$E[f(X,Y)Z].$$ If $f(X,Y)$ and $Z$ are independent then clearly ...
0
votes
0answers
45 views

Deriving conditional variance expression and CDF expansion

Question: Assume a bivariate GARCH process as follows: \begin{align} r_{mt} &= \sigma_{mt}\epsilon_{mt} \ \ \ \cdots \ \ \ \text{(1)} \\ ...
0
votes
1answer
72 views

2 dimensional Brownian motion but not 3 dimensional Brownian motion

Let $W_t = (W_t^{(1)},W_t^{(2)},W_t^{(3)})$ be 3 dimensional Brownian motion. Let $X=sgn(W_1^{(1)})sgn(W_1^{(2)})sgn(W_1^{(3)})$. Define a 3 dimensional process $M_t$ as follows : $M_t^{(1)} = ...
0
votes
2answers
158 views

Conditional Expectation given joint distribution

Given 2 random variables $X,Y$, is it possible to write conditional expectation $\mathbb{E}[X|Y]$ in terms of their joint distributional function $F_{X,Y}(x,y)$?
0
votes
1answer
180 views

Joint Distribution of two correlated ito integral

I have a question regarding finding the joint distribution of two process$$dX_{t}=a_{t}dB_{t}$$$$dY_{t}=b_{t}dW_{t}$$where $B_{t}$ and $W_{t}$ are two Brownian motions with correlated increments, in ...
0
votes
2answers
327 views

finding the distribution of a poisson distribution with random variable lambda

So suppose $X$ is a rv with a Poisson distribution with $\lambda$ being a random variable as well. $\lambda$ has an exponential distribution with mean $1/c$ and $f_\lambda(t) = c\times\exp(-ct)1_{[0, ...
1
vote
2answers
78 views

A question concerning the joint probability distribution

Here is the original question: Given a stochastic process $X(t)=Y_1+tY_2$, where $Y_1,Y_2$ are i.i.d satisfying $Y_1 \sim N(0,1)$. Derive the joint probability distribution for $(X(t),X(s))$ where ...
0
votes
0answers
34 views

Does the spectral gap of an absorbing Markov chain reflect its convergence rate to absorbing state

Spectral gap of a reversible finite-state Markov chain is known to be measure for its rate of convergence to the stationary/limiting distribution. Is this valid for an absorbing Markov chain as well ...
1
vote
1answer
66 views

Covariance combined with normal distribution

We have $N_1$ and $N_2$, normal distributed random variables with averages $µ_i=E[N_i]$ and variances $σ_i^2=Var[N_i]$ and $c = Cov(N_1, N_2)$. We want to compute $E[e^{N_1} I(N_2>0)]$, where I is ...
1
vote
1answer
65 views

Issue with a Poisson process and its jump times

Let $(N_t)_{t\geq 0}$ be a Poisson process and $$T_n = \inf\{t\geq 0, \ N_t \geq n\}$$ Now given $t \ge 0$ how to compute $$ \mathbb{E} \left[ \sum_{n=1}^{N_t} X_{T_n}\right] $$ ? where $(X_t)_{t\ge ...
2
votes
1answer
175 views

Average number of bins occupied when throwing $n$ balls into $N$ bins

There are $n$ balls and $N$ bins. At each time, a ball is thrown in one bin of $N$ bins at random. This repeats n times. So that in total $n$ balls are thrown into bins. The question is, on average, ...
0
votes
1answer
57 views

Sum of strictly stationary sequence

Let $(X_i)_{i\in\mathbb{N}}$ be a sequence strictly stationary real random variables and $k\in\mathbb{N}$. Does \begin{align*} \sum_{i=1}^{n}X_i\end{align*} have the same distribution as ...
0
votes
2answers
257 views

Probability of Poisson event at time t vs. probability of event by time t

I am modeling three events A, B, and C as Poisson processes with rates $\lambda_A$, $\lambda_B$, and $\lambda_C$ and I would like to calculate the likelihood of observing some data given my model. A ...
0
votes
0answers
70 views

Sum of two log-normal

Let X and Y be two GBM’s, they have each a univariate log-normal distribution for some time t, that is $X_t\sim{LnN(µ_x, σ^2_x)}$, $Y_t\sim{LnN(µ_y, σ^2_y})$ and $Z_t=[X_t,Y_t]\sim{ MvLnN(μ, Σ)}$ ...
0
votes
1answer
90 views

First Order Stochastic Dominance

I am reading up on stochastic dominance(http://en.wikipedia.org/wiki/Stochastic_dominance) and have some questions: PDF and CDF of Gamble A and B look like this. Since the CDF of A is always less ...