8
votes
2answers
90 views
+50

How to generate points uniformly distributed on the surface of an ellipsoid?

I am trying to find a way to generate random points uniformly distributed on the surface of an ellipsoid. If it was a sphere there is a neat way of doing it: Generate three $N(0,1)$ variables ...
1
vote
0answers
19 views

Derivative of stochastic process

I have a set of data of a random process (one sample path). The process is sampled every 10 min and each sample is a 10 min average from a sensor. I can compute the statistics of the random process, ...
0
votes
0answers
17 views

Ratio Distribution of Two Dependent Chi-Squared without Joint Distribution

Assume vectors $\textbf{x}$ and $\textbf{y}$ are two independent Complex Gaussian random vectors with i.i.d elements. What is the PDF of $z$ as the ratio of norms in this form: \begin{equation} ...
0
votes
1answer
16 views

Bounds for PDF of Sum of Two Dependent Random Variables

Assume $X$ and $Y$ are two dependent random variables and we do not have the joint distribution of these two. Is there an upper/lower bound for the PDF of $X+Y$? I found a paper which provides bounds ...
2
votes
1answer
32 views

Is this a Markov chain? [duplicate]

Let $\{\xi_n \}_{n \geq 1}$ be i.i.d random variables taking values on $\mathbb{Z}$. Let $\xi_0 = 0$. $S_n = \sum\limits_{i=1}^{n} \xi_i,$ where $S_0=0$ $Y_n = \sum\limits_{i=0}^{n} S_i$. My ...
0
votes
1answer
47 views

Expected value and variance of random process

Let $U,V$ be random variables with distributions $\mathcal{U}(-1,1)$ ,$\mathcal{E}(2)$ (uniform and exponential). If $U$ and $V$ are independent what is the variance and expectation of the random ...
1
vote
1answer
37 views

Distribution of number of Poisson arrivals in interval

$X_1$ and $X_2$ are both Poisson processes. $N$ is the number of arrivals of $X_1$ in between two subsequent arrivals of $X_2$. Derive the probability density $f_N(n)$ of $N$. I wanted to start from ...
1
vote
1answer
36 views

Covariance of a function of random variables

I want to find the covariance $K_X(t,t')$ of the following signal $X(t)$: $X(t)=\sum\limits_{n=-\infty}^{+\infty} A_np(t-nT)$ where $ p(t) = \begin{cases} \ 1 & \text{if } 0<t\leq T/2 ...
3
votes
1answer
71 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
1answer
30 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 ...
-1
votes
1answer
19 views

Finding number of points in a bounded set when number of points in the unbounded set are known.

Consider a random distribution of points in a Random 2D plane. I would like to find the number of points in a circle within this plane. Can anybody helps in solving the problem? Regards
1
vote
1answer
37 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. ...
0
votes
1answer
43 views

continuous RV from discrete RV

So I am reading some notes in stochastic processes and I don't really understand the solution of this problem: Problem: Let $(\Omega,F,\mathbb{P})$ be a probability space where $\Omega$ is the set ...
1
vote
1answer
160 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 ...
4
votes
1answer
103 views

How to find $\mathbb{E}[X\mid\min(X,Y)]$?

Say I have two independent variables $X$ and $Y$ that are exponentially distributed with respective rates $\lambda_X$ and $\lambda_Y$. How do I compute $\mathbb{E}[X\mid \min\{X,Y\}]$?
1
vote
1answer
21 views

Poisson process independence

In a Poisson process with rate $\lambda$ we know that the number of arrivals from $t = 0$ to $t = 1$ is independent of the number arrivals $t = 1$ to $t = 2$. However, are $N(2)$ and $N(1)$ ...
0
votes
1answer
60 views

Conditional probability of random variables

Say I have random variables ~$Exp$ and lets call them $X$ with rate $\lambda$ and $Y$ with rate $\mu$. How do I find $\mathbb{P}\{X>Y|Y>4\}$? I know that $\mathbb{P}\{X>Y\} = \frac{\mu}{\mu + ...
3
votes
1answer
72 views

Kolmogorov's $0-1$ law and constant RV

Kolmogorov's $0-1$ Law: For any terminal event $A$ we have that either $\mathbb{P}(A)=1$ or $\mathbb{P}(A)=0$. Alternatively any $F_{\infty}$ measurable random variable (so basically a terminal ...
-1
votes
2answers
40 views

Gaussian processes are determined by their mean and covariance functions.

A stochastic process $X_t$ is called Gaussian if the random vectors $(X_{t_1},...,X_{t_n})$ are multivariate normal. Why are the finite dimensional distributions of a Gaussian process determined by ...
0
votes
0answers
28 views

Error in thinking: Poisson Process is a Markov Process

I am a bit confused on proving the Markov property for Poisson processes. That is, we want to prove, if $X = (X_t: t \in \mathbb{R})$ is a Poisson process with rate $\lambda$: $P(X_{t_n} = a_n | ...
0
votes
0answers
31 views

linear system output when input is a Gaussian process?

Rectently, I read a technical book that says:" the linear transform of a Guassian process is also a Guassian process. i.e. for continuous time case: $$ x(t)*h(t)=y(t)$$ the input $x(t)$ is a ...
0
votes
1answer
45 views

a question which is somhow related to law of large number

suppose that $\mathbf p = [p_1, p_2, ..., p_n]'$ is a random vector. (' == transpose) and each element of $\mathbf p$ like $p_i$ is a Gaussian random variable with zero mean ($\mathbb E(p_i)=0$) and ...
0
votes
0answers
17 views

Calculating the joint distribution of an affine stochastic process

I have a recursively defined system given by $$X_i = X_{i-1}H_i+N,$$ where $H_i$s are i.i.d. exponential random variables and N is a constant. At the $n$th iteration I have $$X_n = ...
5
votes
1answer
193 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 ...
2
votes
0answers
51 views

Distribution over the time it takes for a random process to reach an upper threshold

I am trying to figure out a way of determining the distribution over the time it takes for an arbitrary random process to cross a threshold value. For example, a simple (solved) case would be the ...
1
vote
0answers
50 views

Hypothesis testing: find the UMP test

Suppose $X_1,\dots,X_n$ are i.i.d. They are distributed as follows: $P(X_i > x)=(1+x)^{-\lambda}$ where $x\geq 0$ and $\lambda> 0$. I have to test the following hypothesis with level $\alpha_0$; ...
1
vote
2answers
389 views

What is difference between stochastic process and a sequence of random variables?

Suppose that I have a sequence of random variables $X=<X_1,...,X_n>$ and I do not have any assumption about these continuous random variables, $X_i$'s, (they can be dependent/independent, ...
1
vote
1answer
114 views

Showing that two Gaussian processes are independent

Say that $Z_t = (X_t, Y_t)$ is a 2-dimensional Gaussian process (by definition, it means that the random vector $(X_{t_1},Y_{t_1},...,X_{t_n},Y_{t_n})$ is a Gaussian random vector for all $t_1 ...
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 $$F_X(x)=\lim_{y \to \infty}F_{XY}(x,y)$$ ...
0
votes
1answer
187 views

What is the difference between a random vector and a stochastic process?

I am a little confused about random vectors and stochastic processes. I read their definitions in Wikipedia (random vector,stochastic process) and I cannot understand their differences . I would ...
3
votes
1answer
146 views

Markov Chain Initial Distribution

Suppose $\{X_0,X_1,X_2,\dots\}$ is a discrete-time Markov chain taking values in a finite set $\{1,\dots,N\}$ with initial distribution $p_i(0) = P(X_0 = i)$ for $i\in\{1,\dots,N\}$ and transition ...
3
votes
0answers
994 views

Relation between standard deviation and mean in random processes

In a Poisson distribution the square of the standard deviation $\sigma$ is equal to mean $\mu$ ($\sigma^2=\mu$) and in a binomial distribution $\sigma ^2=\mu\,(1-p)$ (with $p$ the probability of ...
1
vote
1answer
66 views

Expectation/ independence of random variables

Let $X,Y$ be two correlated variables and $Z\sim N(0,1)$ independent of $X,Y$. Consider the expectation: $$E[f(X,Y)Z].$$ If $f(X,Y)$ and $Z$ are independent then clearly $E[f(X,Y)Z]=E[f(X,Y)]E[Z]=0$ ...
1
vote
2answers
112 views

Is random variable $X_i$ measurable on ${\mathcal F_{i+1}}$ or ${\mathcal F_{i-1}}$?

Let say I have a filtration $\mathcal F_i$ with $\mathcal F_1$ contained in $\mathcal F_2$, $\mathcal F_2$ contained in $\mathcal F_3$ and so on...$\mathcal F_n$. $X_i$ is a stochastic process, $X_i$ ...
0
votes
2answers
118 views

Find the conditional expectation $\mathbb{E}[X_2|\mathcal{F}_1]$

Xn is a sequence of random variable. Let X0 = 4, Xn=2*X(n-1) with prob = 3/4 or Xn=0.5*X(n-1) with prob = 1/4. Fi is a filtration of sigma field. F0 = {null,W} (W stands for omega), F1= sigma (X1) , ...
1
vote
1answer
159 views

Finding the joint distribution of a random process with memory

I'm modeling a digital system as a random process and attempting to solve for the autocorrelation in order to arrive at the power spectral density of the process. The system is as follows: At any ...
0
votes
1answer
130 views

Function of stationary processes

Suppose we have stationary processes $X_1(t), X_2(t),..., X_n(t)$ and let $f_t(X_1(t), X_2(t),..., X_n(t))$ be a continuous function of these stationary processes. Will $f_t(\cdot)$ also be stationary ...
2
votes
1answer
84 views

Given a covarince matrix, generate a Gaussian random variable

Given a $M \times  M$ desired covariance, $R$, and a desired number of sample vectors, $N$ calculate a $N \times M$ Gaussian random vector, $X$. Not really sure what to do here. You can calculate ...
0
votes
1answer
60 views

Given a function and how do you determine the pdf of the left side given the pdf of the right side variables?

Given a function and how do you determine the pdf of the left side given the pdf of the right side variables? Specifically what is the pdf of W, given the equation $$ W = I^2 R$$ with $I$ and $R$ are ...
0
votes
1answer
837 views

Joint pdf of independent randomly uniform variables

Given two random uniform variables, $U$ and $V$, that are uniformly distributed over [0,1], how do you calculate the joint pdf of $X$, $Y$ where $X = F(U,V)$ and $Y = G(U,V)$ and where is the joint ...
1
vote
1answer
82 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 ...
0
votes
1answer
107 views

Probability of Specific event occuring between 2 events?

Forgive me beforehand for what may be a question with an obvious seolution, but I havent had statistics courses in quite some time. I have an Excel File of approximately 3000 Events, each event has a ...
2
votes
1answer
123 views

Weighted integral of random variables

Given a random zero-mean gaussian random variable $X(t)$ with parameter $t$, such that $E [X(t) X(t^\prime)] = \sigma^2 (t) \delta_{tt^\prime}$, is it possible to produce a single gaussian random ...
2
votes
0answers
93 views

Absolute Continuity and simple discontinuity

I am reading a book called Stochastic Process, Estimation, and Control, in P.32 it states that a function with finite simple discontinuities can still be absolutely continuous, which confused me, I ...
2
votes
3answers
233 views

Proof of Levy's zero-one law

Let $(\Omega, \mathcal{F},\mathbb P)$ be a probability space and let $X$ be a random variable in $L^1$. Let $(\mathcal{F}_k)_k$ be any filtration, and define $\mathcal{F}_{\infty}$ to be the minimal ...
0
votes
1answer
141 views

Examples of convergence of random variables

First, let's recall the definitions of 4 different types of convergence:almost surely, in $r$th mean, in probability and in distribution: $X_n\xrightarrow{a.s.}X$ if $\{\omega \in ...
1
vote
1answer
31 views

Is $\{ r \mapsto X_{r} \text{ is continuous for all } s < t \} \in \sigma(X_s : s \leq t)$?

If $(X_t)_{t \geq 0}$ is a stochastic process, is $\{ r \mapsto X_{r} \text{ is continuous for all } $s < t$ \} \in \sigma(X_s : s \leq t)$? I'm particularly interested in the case where $X_t$ is ...
2
votes
2answers
291 views

First jump time of Poisson process (and general right-continuous processes).

I've read that the first jump time of the Poisson process is totally inaccessible (definition at the bottom for anyone interested). This made me wonder if the first jump time is a stopping time. I ...
1
vote
0answers
77 views

Showing a certain process has $\limsup X_t$ bounded almost surely.

This question has been solved. I'm working on a problem where I need to show $$\limsup_{t \rightarrow \infty} X_t \leq \sqrt{c}\quad \text{a.s.}$$ where $X_t$, $t \geq 0$ is a stochastic process ...
1
vote
1answer
58 views

Inequality between two Random Walks

Let's consider two Random Walks, $$x^{(1)}_t = x_0 + \sum_{i=1}^{t}\xi^{(1)}_i,$$ $$x^{(2)}_t = x_0 + \sum_{i=1}^{t}\xi^{(2)}_i.$$ The random variables $\xi^{(1)}_i$ are i. i. d. They take values on ...