Tagged Questions

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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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: $P\{X = i \} = p(1-p)^{i-1} , i \geq 1$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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How can I determine this equation?

I have an equation that have been formulated as: $$\Delta w_{ki}=\eta (y_{k}-y_{o})(x_i-x_o)+a_1,$$ where$$y_{k}=\sum_{j}w_{kj}x_{j}+a_{2}$$ and where $a_1,\eta,x_o ,y_o$ and $a_2$ are constants and ...
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Adaptation of sum of arrival times of Poisson process

Let $\{N_t\}_{t\geq0}$ be a Poisson process and $\{F_t\}_{t\geq0}$ be its nautral filtration so that $\{N_t\}_{t\geq0}$ is adapted. $T_i$ be the $i$th arrival time of Poisson process of arrival rate ...
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Expectation of sum of arrival times of Poisson process in $[0, t]$

Let $T_i$ be the $i$th arrival time of Poisson process of arrival rate $\lambda$, given $t>0$, how to calculate $$E(\sum_{i=1}^\infty T_i 1_{\{T_i<t\}})$$ I think since this is equal to  ...
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Why Gibbs sampling needn't “remixing”

I am generating $\mathbf{x}^{(1)}, \mathbf{x}^{(2)}, \dots, \mathbf{x}^{(n)}$ using Gibbs sampling methods. So I want $\mathbf{x}^{(1)}, \mathbf{x}^{(2)}, \dots, \mathbf{x}^{(n)} \sim$ some ...
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Covariance of *sum* of k autocorrelated variables

This problem was part of the preliminaries required to answer a question in an assignment set for a subject I'm doing: I've been scratching my head about it for two days now (having submitted the ...
Suppose you have the history of some finite initial segment of a Galton–Watson branching process whose offspring distribution is a Poisson distribution with expectation $\mu$. By "initial segment" I ...