# Tagged Questions

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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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### Deducing an optimal gambling strategy (using martingales).

Apologies in advance for the length, I tried being precise. Suppose a game where in each turn you can gamble a certain amount of money on the result of a fair coin toss. If the coin comes out tails ...
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### Filling of a tank - recurrence relation

Suppose a tank has a maximum limit of 100 units. Each day 2,1 and 0 units are added to the water level with probability p,r and q. Any excess water would overflow and if it reaches the minimum level ...
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### Need help with a basic exercise about Markov chains

Suppose $\left\{ X_{n}\right\} _{n=1}^{\infty}$ is a Markov Chain taking real values. Are the following Markov Chains? $$Y_{n}=\sum_{i=1}^{n}X_{i} , Z_{n}=\left(X_{n},X_{n-1}\right)$$ Edit1 I ...
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### Random process with Cauchy distribution

The problem is as follows. Let $X(t)$ be a stochastic process such that $X(t) = V + 2t, t \ge 0$, and $V$ has the Cauchy distribution $x_0 = 0, \gamma = 1$. Find the probability that $X(t) = 0$ for ...
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### A question on recurrent events

In a sequence of Bernoulli trials let E occur when the accumulated number of successes equal to $c$ times the number of failures where $c$ is a positive integer. I need to show that E is persistent if ...
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### Discrete Time Markov Chain - Bank customers

Suppose $N_n$, the number of customers arriving at a bank during day $n$, is distributed Binomial$(p, m)$. Consider the simple situation where each customer either withdraws £1 or deposits £1 with ...
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### Finding the probability of occurrence of one event before an another

In a home work problem $E$ and $F$ are mutually exclusive events in the sample space of an experiment. The experiment is repeated until either event $E$ or event $F$ occurs. Show that the probability ...
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### Poisson Process (Easy)

I'm stuck at the following question: Customers with items to repair arrive at a repair facility according to a Poisson process with rate λ. The repair time of an item has a uniform distribution on ...
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### Almost sure non differentiability of Brownian Motion

Problem: Let $t>0$, show that the standard Brownian motion is almost surely not differentiable a $t$ Now, through a Borel Cantelli argument I proved that, almost surely \limsup_{\epsilon ...
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### period of product markov chain

Consider $Z_n := (X_n,Y_n)$ where $(X_n)_{n\in \mathbb{N}}$ and $(Y_n)_{n\in \mathbb{N}}$ are irreducible markov chains with periods $\lambda$ and $\mu$. We know that $(Z_n)_{n\in \mathbb{N}}$ is a ...
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### Variance of drawing coins from a bag.

First off, disclaimer, this was a homework question, albeit one that I've already turned in. I was given the problem ...
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### Local martingale is locally uniformly integrable martingale?

Is a local martingale locally uniformly integrable martingale ? Here I define a local martingale to be the process with a localizing sequence $\tau_n$ such that the stopped process is martingale. ...
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### simplified iid Galton-Watson-process — expectation and variance of population size

Let $Y_{ni}$ be iid and take on values in $\{0,1,2,\ldots\}$. Set $Z_0=1$ and define $Z_n:=\sum_{i=1}^{Z_{n-1}}Y_{ni}$ where by convention the sum is zero if $Z_{n-1}=0$. Let $E(Y_{11}) = \mu$ and ...
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### Is this two dimensional Markov chain correct for this queueing system?

The problem that I have two single server station with no queuing space a customer goes to station 1 if it is available else it goes to station 2 if it is available or it will be lost output from ...
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### Analyze the packet loss rate

Suppose that node A send $m(i)$ ($m(i)$ obeys i.i.d. poisson distribution with parameter $\lambda$) packets to node B at time slot $i$, and B can handle $C$ packets at each time slot. There is a ...
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### Stochastic dynamic programming

I am making some homework exercises at the moment and I was wondering if what I did in the following exercise was correct. PROBLEM Solve $E(\sum_{k=0}^{N-1}(1-u_k)X_k + X_N) \rightarrow \max$, ...
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### Compute $d(\log(S_t))$ using Ito's Formula

We are given the following: d$S_t$ = $\sin(S_t)t^2dt + e^{\sqrt{S_t}-t}dB_t$ And are asked to compute several different things, one of which is $d \log(S_t).$ If I'm understanding Ito's formula ...
The question is if $x,y,z$ are independent $x\sim\exp(\lambda), y\sim\exp(\mu), z\sim\exp(\gamma)$ and define $u=\min(x,y), v=\min(y,z)$ what is the probability $p(U>u,V>v)$. Consider the cases ...
Suppose we have a branching process, where at each time $n$, each individual produces offspring independently with the distribution $\{p_k\}$ and then dies with probability $0 < q < 1$. For ...