# Tagged Questions

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### Using the Probability generating function to find the probability of ultimate extinction

I am having problems with an exam question from a past paper, help would be appreciated: Let $X_n$ be the number of carriers of a family name in the nth generation and suppose $X_0=a$. Suppose ...
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### Compute a probability using stochastic simulation

My problem is to solve the following optimisation problem using GA (Genetic Algorithm)and stochastic simulation. The goal is to solve the maximisation problem : \begin{equation*} \begin{aligned} ...
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### I want to show $e^{-\alpha t}B(e^{2\alpha t})$ is a Gaussian process and I find mean and covariance functions

Let $B(t)$ be Brownian motion. Show that $e^{-\alpha t}B(e^{2\alpha t})$ is a Gaussian process. Find its mean and covariance functions. thanks .
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### I want to calculate $\int B(t)^2 dB(t)$ where $B(t)$ is Brownian motion

Let $B(t)$ be Brownian motion. I want to calculate $\int B(t)^2 dB(t)$. definition.A process $\{X(t),0\le t \le T \}$ is called a simple adapted process if there exist times ...
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### I want to show some properties for Brownian motion

Let $B(t)$ be a Brownian motion. Show that the following processes are Brownian motions on $[0,T]$ 1) $X(t)=-B(t)$; 2) $X(t)=B(T-t)-B(T)$, where $T\lt \infty$; 3) ...
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### How to prove $(X_{n})_{n\in \mathbb N}$ and $(Y_{n})_{n\in \mathbb N}$ are supermartingale and $(Y_{n})_{n\in \mathbb N}$ is convergence to -7

Let $p \in [0 , \frac{1}{2}]$ and $\eta_{i}$ be i.i.d random variables and $P(\eta_{i}=1)=p$ and $P(\eta_{i}=-1)=1-p$ and $\mathcal F_{n}=\sigma(\eta_{1},\cdots,\eta_{n})$ and ...
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### What is the expected number of steps before the spider exits? Markov Chain

A little spider lives in a rectangular box of which the sides are 3 and 4 cm long. It can only sit in one of the four corners marked with the numbers 1,2,3,4 (clockwise). Assume in Corner 2 there is a ...
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### Show that $T=\min\{n:X_{n}\in B\}$ is an $\mathcal F_{n}$-stopping time

Let $X_{n}$ be an $\mathcal F_{n}$-martingale and let $B\in \mathcal B$. Show that $T=\min\{n:X_{n}\in B\}$ is an $\mathcal F_{n}$-stopping time. $\mathcal B$ is Borel $\sigma$-algebra and filtration ...
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### Is $X_{i}$ a martingale, submartingale, or supermartingale?

Let $X_{i}$, $i=0,1,\cdots$ be a sequence of random variables generated by $X_{i+1}=\rho X_{i}+W_{i}$, where $\rho$ is constant and $W_{i}$ are i.i.d random variables. Suppose $X_{0}$ is independent ...
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### If $X_{n+1}$ is a martingale subject to $Y_0,\ldots,Y_n$, then is it a martingale with respect to $Y_0^2,\ldots,Y_n^2$?

I don't have a very solid foundation in measure theory, and this always seems a bit confusing to me so I would appreciate any help. We are given $E \left( X_{n+1} | Y_0,\ldots,Y_n \right) = X_n.$ ...
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### How to derive the conditional variance and CDF of a bivariate GARCH process?

Question: Assume a bivariate GARCH process as follows: \begin{align} r_{mt} &= \sigma_{mt}\epsilon_{mt} \ \ \ \cdots \ \ \ \text{(1)} \\ ...
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### Error metrics to assess a stochastic model with Monte Carlo simulations

Suppose we have constructed a model of some stochastic system; we are also able to perform Monte Carlo simulations of this system. Now, we have two sets of samples: one from our model and one from the ...
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### Expected number of coin tosses to get N consecutive, given M consecutive

Interviewstreet had their second CodeSprint in January that included the question below. The programmatic answer is posted but doesn't include a statistical explanation. (You can see the original ...
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### Taking random items out of a container with replacement

I take vitamins in the morning, but one of them I only take a half tablet. So, I have an initial container with 100 full tablets, and every morning I take out a random tablet. If it's a full ...
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### Moments of function of Poisson process

Mirror thread on Mathoverflow. (I'm new to Poisson processes, so please edit if my terminology is incorrect.) This is a special case of a problem I'm working on; hoping for intuition that will ...
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### Time taken to hit a pattern of heads and tails in a series of coin-tosses

Inspired by Peter Donnelly's talk at TED, in which he discusses how long it would take for a certain pattern to appear in a series of coin tosses, I created the following script in R. Given two ...
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### Conditions for Central Limit Theorem for dependent sequences

Cumbersome technical assumptions (e.g., mixing properties) are used in the literature to prove Central Limit Theorems for dependent sequences. I sketched a proof that does not require any of these ...
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### Steady state probabilities for a continuous-time Markov chain

I have a finite state and time-homogeneous continuous-time Markov chain (CTMC) which is not irreducible. Will steady state probabilities exist for this CTMC? How to prove this?
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### Meaning of this expectation equation?

I was actually looking at this problem on slide 12. I will write it here briefly: Problem: Unknown number of people arriving in a fixed time period and my goal is to maximize my probability of ...
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### Origin of strange formula for equilibrium standard deviation

In the paper M. Avellaneda and J. H. Lee, Statistical arbitrage in the U.S. equities market, July 2008, in the Appendix on page 46, how does he get equilibrium standard deviation as following: ...
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### Why local martingale property is important in Girsanov theorem?

In Girsanov theorem, the change of probability measure variable $Z_t = \frac{dQ}{dP}|_{\mathcal{F}_t}$, why does it need to be a martingale with respect to measure $P$ for the change of measure ...
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### How will studying “stochastic processes” help me as a statistician?

I wish to decide if I should take a course called "INTRODUCTION TO STOCHASTIC PROCESSES" which will be held next semester in my University. I asked the lecturer how studying such a course would help ...
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### Trees generated by multi-type branching processes in n steps

I am trying to develop some algorithm to compute probabilities in multi-type branching trees, and I doubt I am doing this right... Let us consider a multi-type branching process with two types, ...
### What is the distribution of $\chi^n_k$?
$\chi^n_k=\sum_{i=1}^kx_i^n$ where $x_i$ are Gaussian variables and $n>2$?