0
votes
0answers
18 views

Jeffrey's Prior for Bivariate Lognormal

Exactly what the question says, I'm working on code for an MCMC simulation and need to set some uninformative or weakly informative priors. I haven't been able to find the prior for the sigma ...
0
votes
0answers
19 views

kullback liebler divergence for correlated processes

Suppose $X_n^{(1)}=\lambda_1 X_{n-1}^{(1)}+\mu_1+\epsilon_n^{(1)}$ and $X_{n}^{(2)}=\lambda_2X_{n-1}^{(2)}+\mu_2+\epsilon_n^{(2)}$ where $|\lambda_i|<1$ for $i=1,2$ and $\epsilon_n^{(i)}$ are ...
0
votes
0answers
11 views

Determining Causal Relationships Between Two Processes (Pierce-Haugh)

Pierce and Haugh did some research on causality in temporal systems. For simplicity, consider two time series $\{X_{t}\}$ and $\{Y_{t}\}$. Suppose that both follow a causal and invertible $ARMA$ ...
0
votes
0answers
14 views

Asymptotic Distribution of Cross-Correlation Function Between Two AR(1) Processes

Let $$X_{t} = \phi_{x}X_{t-1} + e_{t}$$ and $$Y_{t} = \phi_{y}Y_{t-1} + \epsilon_{t}$$ be two $AR(1)$ processes. My textbook writes that $\sqrt{n} \cdot r_{XY}(k) \sim N \left(0, \frac{1 + ...
0
votes
0answers
10 views

Determining Stationarity of (Multivariate) Processes

A necessary and sufficient condition for a vector autoregressive process to be stationary is for its eigenvalues to all have modulus less than one. I learned this result in the case of a model of this ...
1
vote
0answers
13 views

Fitting of the Lévy triple

Given a Lévy process and its triplet $(\mu,\Sigma,\nu)$ i.e. the triplet such that for each $t\ge 0$ $ X(t) = bt + W_A(t) + \int_{|x|<1} x \tilde N (t, dx) + \int_{|x|\ge 1} x N(t,dx)$ where ...
0
votes
1answer
20 views

Autocorrelation of a Wiener Process proof

Given a Wiener process X, how do I prove this? $R_x(s,t) = E[X(s)X(t)] = min(s,t)$ There seems to be a trick with dividing to two cases of $s<t$ and $s>t$, but I can't figure out how this ...
0
votes
0answers
30 views

Autocovariance Function of this $AR(1)$ Process

Consider the $AR(1)$ process given by $(1-0.6B)(X_{t} - 3) = a_{t}$ where $a_{t} \sim WN(0,1)$ and $B$ is the backshift operator ($X_{t}B = X_{t-1}$). We can rewrite the process in the more ...
1
vote
1answer
58 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 ...
1
vote
1answer
27 views

Determining Moving-Average Representation of AR(2) Process

Consider a stationary $AR(2)$ process given by $$X_{t} - X_{t-1} + 0.25X_{t-2} = 5 + a_{t}$$ where $a_{t} \sim WN(0,1)$ (white noise). I am interested in obtaining the causal representation of ...
0
votes
0answers
29 views

Matlab code for higher order scheme

Can somebody help me how to generate the code for the increment $\Delta$Z in the document I have attached? I know how to generate the rest of the increments but struggling in how to generate ...
1
vote
1answer
23 views

Likelihood function of a Poisson process

Fix a window of time $[0,T]$ and say that we get $n$ arrival times in the window from a homogeneous Poisson process. The maximum likelihood estimate (MLE) is just $n/T$ I believe. But what is the ...
1
vote
0answers
30 views

Definition of Time Series

Having not done any stats for a few years, I seek clarification regarding the definition of time series given in my textbook. I apologize for the length, but I would be glad to just resolve my main ...
-1
votes
1answer
27 views

Two Reflecting Barriers

A chain with stats 1,2,....,n has a matrix whose first and last rows are (q,p,0,...,0) and (0,...,0,q,p). In all other rows Pk,k+1 = p, Pk,k-1 = q. Find the stationary distribution. I am ...
0
votes
0answers
29 views

independence at equal and different times

this is a question about stochastic processes. Let's call $A(t)$ and $B(t)$ two stationary processes and denote by $E[*]$ the expectation value. Suppose we know that $E[A(t)B(t)]=0$ for every $t$. The ...
0
votes
0answers
48 views

Poisson/ jump process distribution for process $z(t)=2t+B(t)+\sum_{k=0}^{X(t)} J_k$

For the process: $z(t)=2t+B(t)+\sum_{k=0}^{X(t)} J_k$, where $X(t)$ is a poisson process with paramater $\lambda$, and: $J_k$ are i.i.d . random variables (jumps). $B(t)$=brownian motion. I want to ...
1
vote
1answer
42 views

Literature on Sabermetrics in baseball

For my bachelor's thesis, I would like to study the use of Sabermetrics in baseball. I was fascinated by the book 'Moneyball: The Art of Winning an Unfair Game' by Michael Lewis, and to me, it ...
1
vote
1answer
24 views

questions on a property of ARCH model

When reading the book of Analysis of Financial Time Series, I have a question on the ARCH model, defined as follows Regarding this model, the author also states ...
0
votes
0answers
28 views

Poisson process different type of events

Suppose that it arrives people to a store according to a poisson process with rate $\lambda = 6$/hour , females arrive with probability $0.6$ and male with $0.4$. What is the probability that there ...
0
votes
1answer
30 views

Justification of Poisson postulates

This may be a dumb question. The Poisson postulates are: $P(n=1,h) = \lambda h + o(h)$ $\sum\limits_{i=2}^{\infty}P(n=i,h) = o(h)$ Events in nonoverlapping intervals are independent What ensures ...
0
votes
1answer
18 views

Joint density of order statistics

I need some help to undertand the following proposition (mainly to understand how they prove it): Let $Y_1,Y_2...,Y_n$ be $n$ random variables which are independent ,identically distributed random ...
1
vote
2answers
32 views

Poisson process counting process

Two individuals, A and B, both require kidney transplants. If she does not receive a new kidney, then A will die after an exponential time with rate $\mu_A$, and B after an exponential time with rate ...
1
vote
1answer
34 views

Ehrenfest urn model expectation question

Consider the Ehrenfest urn model in which $M$ molecules are distributed between two urns, and at each time point one of the molecules is chosen at random and is then removed from its urn and placed in ...
1
vote
1answer
19 views

expectation by reasoning

An unbiased die is successively rolled. Let $X$ and $Y$ denote, respectively, the number of rolls necessary to obtain a six and a five. $E[X]= 6$. find $E[X \mid Y=1]$ Iam stuck on this. Iam ...
0
votes
0answers
17 views

Autocorrelation equality of multiple processes

Given: I have 2+ time series and theirs ACF. Situation I need some criteria of ACF equality for all of them, I mean $\forall i, ACF_i(t)\equiv F(t)$. I wanted to use least squares, but my ...
3
votes
1answer
55 views

A linear growth model with immigration

Ill give some background first before asking questions.(the text below is straight out of the book) Each individual in the population is assumed give birth at an exponential rate of $\lambda$ in ...
1
vote
1answer
20 views

Simulate random variable RV

i need some help to understand what is going on here.I'll be more specific on questions after i give the context. Suppose we want to simulate $X$ such that: $P\{X = i \} = p(1-p)^{i-1} , i \geq 1 $ ...
1
vote
1answer
85 views

Mean time spent in transient states/Markov chain

I dont get this in my book: For transient states $i$ and $j$ , let $s_{ij}$ denote the expected number of time periods that the markov chain is in state $j$ , given that it starts in state $i$. Let ...
0
votes
0answers
24 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 ...
1
vote
1answer
101 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 ...
0
votes
1answer
38 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 ...
2
votes
2answers
158 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
1answer
51 views

Similarity between two curves

I am trying to find out how well a deterministic version of a MATLAB program predicts the stochastic version. I don't know what statistical test/quantitative analysis to use. This is what the output ...
0
votes
0answers
7 views

How can I calculate distribution of minima of sections of a continuous path (from a stochastic process)?

I have a long slab whose width is defined by a stochastic process, whose complete statistics I am aware of, say. I now cut it into smaller sections of uniform length, and calculate the minimum width ...
1
vote
0answers
34 views

Expected Sum of Weights after Drawing Without Replacement

We have an urn containing $k$ balls where for all $i:1\le i\le k$, the ball $b_i$ has the size $s_i$ that determines its probability to be drawn. For instance, a ball $b_i$ with size $s_i=3$ is ...
3
votes
1answer
88 views

I have trouble understanding the proof of the Wold decomposition theorem

I'm trying to understand the proof of the Wold decomposition theorem in [1, p.187]. I find a few things about it very irritating. The theorem states: Theorem 5.7.1 (The Wold Decomposition). Let ...
0
votes
1answer
38 views

Markov Chain Conditional Probability

A Markov chain has the transition probability matrix as follows. $$To$$ $$ From \begin{matrix} STATES& 0 & 1 & 2 \\ 0 & 0.6 & 0.3 & 0.1 \\ 1 ...
1
vote
4answers
69 views

Displacement law: $ \sum_{i=1}^{n} (x_i - \bar{x})^2 $ why is it squared?

$ \sum_{i=1}^{n} (x_i - \bar{x})^2 $ where $ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i $ (mean value) I would like to understand why the term inside the sum gets squared. Why isn't it enough to e.g. ...
1
vote
1answer
81 views

How to get Expectation and Variance of Geometric Distributions

A random variable N follows a geometric distribution with the "Success Rate" = $\alpha$, so that $P [N=k] = \alpha (1-\alpha)^{k-1},$ for $k$ = 1,2,... Random variables $(Y_1 , Y_2...)$ are ...
0
votes
0answers
41 views

Expectation, Variance, and Value of a Constant

I think I have everything right, I would like someone to check my work. A random variable X has the following density function (Beta-Distribution) (Density) = f(x) = Cx^2 (1-x)^2 (If 0 < x < 1 ) ...
1
vote
1answer
61 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} ...
0
votes
0answers
32 views

How can I conceptually visualize a Gaussian Mixture Model?

I'm struggling to get my head around Gaussian Mixture Models & Processes from a conceptual level; my intention is to use a GMM for image segmentation but I'm struggling to understand it. The way ...
2
votes
1answer
55 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 ...
3
votes
4answers
49 views

Reference request for stochastic process and applications

I am looking for a text book that will cover the following topics I hope someone could suggest me a good text book that will provide me a good guidance regarding the following; Generating functions, ...
2
votes
2answers
38 views

a probability question related to computing the variance of a specific pattern

With respect to a given sequence of points $\{X_1, ... X_t, ...X_n\}$. I can understand why $E[S]= \frac{n-1}{2}$. But how to get that $Var[S]$.
0
votes
0answers
57 views

Poisson process and the Uniform distribution

For any $t>0$ and $n=1,2,...$ is that $$P\lbrace S_{k}\leq x\mid N(t)=n\rbrace =\sum_{j=k}^{n} \binom {n} {j}\left(\dfrac{x}{t}\right)^{j}\left(1-\dfrac{x}{t}\right)^{n-j} $$ for $0\leq x\leq t$ ...
1
vote
0answers
64 views

Computing Replacement rate in a renewal process

I found this problem on some slides that I'm studying as I prepare for a final: "A machine in use is replaced by a new machine either when it fails, or when it reaches the age of $T$ years. If the ...
1
vote
0answers
89 views

Expectation of $n$-dimensional Inverse Bessel Process

I think the main problem for me is to calculate the integral of $$\int_{0}^{\infty}\frac{e^{-\frac{r^2}{2t}}}{\sqrt{x^2+r^2}}r^{n-1}dr,n\geq2$$ For n=2, change of variable $y=\sqrt{x^2+r^2}$ would ...
0
votes
1answer
169 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 ...
2
votes
2answers
97 views